A Numerical Study of Transport Phenomena in Porous Media
by
May-Fun Liou
Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy
Thesis Adviser: Dr. Isaac Greber
Department of Mechanical and Aerospace Engineering
Case Western Reserve University
August, 2005
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______
candidate for the Ph.D. degree *.
(signed)______(chair of the committee)
______
______
______
______
______
(date) ______
*We also certify that written approval has been obtained for any proprietary material contained therein.
For my beloved parents,
Bow-Churng
&
Chung Wan
ii Table of Contents
List of Tables vii
List of Figures viii
Nomenclature xvi
Abstract xx
1 Introduction 1
1.1 General Overview 2
1.2 Volume Averaging Method 8
1.3 Three Dimensional Navier-Stokes Equations 13
1.3.1 Fluid-phase Equations 13
1.3.2 Solid-phase Equation 15
1.4 Material under Studied 15
1.5 Outline of the Thesis 16
2 An Enabling Technique for Pore-scale Modeling 18
2.1 Introduction 18
2.2 Mesh-based Microstructure Representing Algorithm 22
2.2.1 Mesh Generator 28
2.2.2 Random Number Generator 32
2.2.3 Quality Control 34
2.2.4 Porosity 37
2.3 Formation of the Two - dimensional Geometry 38
iii 2.4 Formation of the Three - dimensional Geometry 40
2.5 Numerical Solutions 43
2.5.1 Governing equations - both fluid and solid phases 43
2.5.2 Boundary conditions 44
2.5.3 Numerical methods 46
2.6 Sensitivity Test 47
2.6.1 Geometric types 47
2.6.2 Grid dependence study 47
2.6.3 Convergence study in terms of residual 49
2.7 Discussion 50
2.8 Summary 51
3 Fluid Flow and Pressure Drop in Porous Media 53
3.1 Introduction 53
3.2 Validation Cases 56
3.2.1 Laminar flow over a square cylinder 56
3.2.2 Laminar flow over a circular cylinder 57
3.3 Numerical Experiments of 2D Porous Flow 60
3.4 Results and Discussion 61
3.4.1 Two-dimensional channel flow 62
3.4.1.1 Velocity field 63
3.4.1.2 Pressure drop and porosity 70
3.5 Summary 73
iv
4 Forced Convection Heat Transfer in Porous Media 74
4.1 Introduction 74
4.2 Validation Cases – forced convection in single plate-plate channel 84
4.2.1 Bottom plate as an inner plate 86
4.2.2 Bottom plate as an external plate 90
4.3 Numerical Setup for Forced Convective Heat Transfer 94
4.3.1 Two-dimensional channel flow 95
4.3.1.1 Results and discussion 97
4.3.1.2 Effective thermal conductivity 117
4.3.1.3 Thermal dispersion 122
4.3.2 Summary 137
4.4 Three-dimensional Rectangular Duct Flow 138
4.4.1 Flow with Heated Porous Section 138
4.4.2 Summary 150
4.5 Conclusion 150
5 Closure
5.1 Summary 152
5.2 Future Work 154
Appendix A Programs of Random number generating 156
v Appendix B Governing equations and numerical methods 160
Governing equations 160
Numerical method 162
Appendix C Fluid flow and pressure drop data 168
References 169
vi List of Tables
Table 1 : List of thermophysical properties of solid materials under study……… 16
Table 2 : Results from grid-dependence study………………………………….. 48
Table 3 : Summary of previous works in evaluating viscous resistance (pressure drop) in porous media……………………………………………………….. 54
Table 4 : List of pressure drops for various porous systems…………………….. 61
Table 5 : Summary of numerical simulations of forced convective heat transfer in Porous media………………………………………………………….. 80
Table 6 : The effective heat conductivity of fluid phase at Re = 270…………… 120
Table 7 : The ratio of effective heat conductivity of fluid phase, f, at Re = 1,100. 121
Table 8 : The ratio of effective heat conductivity of fluid phase, f, at Re = 2,200.. 122
Table C-1: The pressure drop of a simplified electronic cooling system at various conditions………………………………………………………………. 168
vii List of Figures
1.1 Schematic of porous medium, flow variations in the representative Elementary
Volume and the pore velocity vector Vp...... 8
2.1 A Schematic explaining the procedures of generating a numerical porous
sample………………………...... 28
2.2 A porous sample with non-uniform porosity……………………………… 34
2.3 Schematic of typical domain of a porous medium – two phase problem…. 37
2.4 Randomly formed solid matrix representing porous medium in 2-D domain
(a) fibrous shape formed with triangular mesh,
(b) small solids in circular shape formed with triangular mesh,
(c) large solids in circular shape with hybrid mesh……………………… 39
2.5 Open-cell foam microstructure …………………………………………… 41
2.6 (a) the cross-section of a typical mesh-based porous medium is shown
with solid balls in black and space taken by fluid in white……………. 41
2.6 (b) The cross-section of numerical generated open-cellular sample after
converting all solid and fluid tags from Figure 2.8(a), into the oppose
ones……………………………………………………………………... 42
2.6 (c) 3D Numerical fluid and solid mesh in cross cut planes along the axial
direction – x direction. …………………..……………………………… 42
viii 2.7 Convergence history of residuals from solving Navier-Stokes equations for
fluid and solving heat conduction equations for solid in the case of the channel
flow heated from both top and bottom walls. …………………...………… 50
3.1 Streamlines of a low Reynolds number flow exhibits symmetry around the
solid block ………………………………………………………………… 57
3.2 A hybrid mesh is used for studying unsteady laminar flow over a stationary
circular cylinder. ………...………………………………………………… 58
3.3 Contours of u velocity indicate that the laminar flow is fully developed in the
flow field …..……………………………………………………………… 59
3.4 Schematic numerical setup for measuring pressure drop …………………... 60
3.5(a) The velocity distribution along the distance from the bottom wall, y', with
respect to sampling locations of ∆1, ∆2, ∆3, ∆4, ∆5 and at the exit of the
channel …...………………………………………………………………. 64
3.5(b) The velocity distributions in the axial (X) and vertical direction (Y)
respectively ..………… ..………………… ……………………………… 65
3.6(a) Velocity contours of the whole numerical domain for the inflow velocity
0.1m/s ………………………..……………………………………….... 66
3.6(b) Vortices contours of the whole numerical domain for the inflow velocity
0.1m/s …………………………………………………………………… 67
3.7 (a) The pressure distribution along the non-dimensional axial direction with
respect to several y locations close to the wall……………………. …… 68
3.7 (b) The pressure distribution along the non-dimensional axial direction
at three y locations, farther away from the wall………………………... 69
ix ∗ 3.8 Dimensionless pressure gradient parameter, P vs. Red, of various porous
systems in Table 3 ………………………………………………………….. 70
∗ 3.9 Non-dimensional pressure drop P vs. Red in a wide range of Reynolds
numbers for two values of porosities.……………………………………… 71
∗ 3.10 A close-up view of the non-dimensional pressure drop P vs. Red, exhibiting
different behaviors caused by varying porosityε …………………………. 72
4.1 Schematic cooling of electronic board with a substrate between coolant and
board. ………………………………………………………………………. 85
4.2 Silicon rubber substrate with isothermal walls at 900 K, (a) velocity contours,
(b) temperature contours…………………………………….……………… 87
4.3 Local Nusselt number Nux versus dimensionless x of heat transfer at the contact
surface of coolant and substrate of silicon rubber………………………. 88
4.4 Aluminum substrate with constant wall temperature, Tw=900K, (a) velocity
contours, (b) temperature contours……………………………………… 89
4.5 Silicon rubber substrate with Tw = 900 K and the other parallel plate insulated
(a) velocity contours, (b) temperature contours………………………… 90
4.6 Local Nusselt number Nux versus dimensionless x of heat transfer at the
contact surface of coolant and substrate of aluminum………………… 92
4.7 Aluminum substrate with isothermal top wall at Tw = 900 K, and the lower
parallel plate insulated: (a) velocity contours, (b) temperature contours. 93
4.8 Nux along longitudinal direction x at the contact surface of coolant and
substrates of silicon rubber and aluminum…………………………….. 94
x 4.9 Schematic of a two-dimensional porous channel with heat source from
below or above…………………………………………………………. 95
4.10(a) Temperature contours of the nonporous channel with the background
showing the mesh used………………………………………………… 99
4.10(b) The invariant temperature profiles along the transverse direction in the
channel at various sampling positions along the axial direction………. 99
4.10(c) Velocity profiles along the transverse direction at all sampling positions.100
4.10(d) Local Nusselt number Nux versus the longitudinal coordinate x of the
baseline case…………………………………………………………… 100
4.11(a) Temperature contours of the porous sample in a channel, along with the
mesh used and the porous region in which the solid matrices are shown in
black…………………………………………………………………… 101
4.11(b) The invariant temperature profiles along the direction of y* in the channel
are plotted for various sampling positions……………………………. 102
4.11(c) Profiles of non-dimensional x-component velocity along the transverse
direction at various sampling locations………………………………… 104
4.11(d) Plot of local Nusselt number Nux versus the longitudinal coordinate x… 104
4.12 The ratio, f, of local heat conductivity of fluid phase to that at the inflow
condition (300K and 1atm) in terms of the distance away from wall at
various sampling locations……………………………………………… 105
4 .13 Distributions of local dimensionless heat transfer along the bottom wall of
four solid phases, the upward spikes indicate the enhancement of heat
transfer contributed from the physical blockages of porous media……. 108
xi 4.14 Vorticity distribution along the dimensionless transverse direction, y, on the
mid-plane of the porous medium (∆3)…………………………………. 109
4.15 Vorticity distributions on the center plane of the hypothetical solid/air
(system of θ = 1.0) under Case (b)……………………………………... 111
4.16 Local dimensionless heat transfer rate along the bottom plate at various
Reynolds numbers. The porous system studied is composed of air and the
hypothetical solid/air as the solid phase, i.e. θ =1……………………… 112
4.17 (a) - (c) The local Prandtl number profiles along the distance away from the
heated wall at various x* locations, where — 0.25 (∆1), — 0.45 (∆2), and —
0.5 (∆3) — 0.55 (∆4) , and — 0.75 (∆5), of (a) Re = 1,100 (b) Re = 2,200 and
(c) Re = 5,500…………………………………………………………… 114
4.18 (a) - (e): Profiles of effective heat conductivity at five planes, ∆1-∆5,
respectively shown in (a)-(e), for three Re=1,100 (─), 5,500 (─), and 11,000
(─)………………………………………………………………………. 115
4.19 Local Nusselt number along the bottom wall, showing the effects of Re, heat
conductivity and specific heat of solid phases…………………………. 116
4.20 Profiles of ratio of heat conductivities f in the transverse direction with
respect to two inflow conditions (Re)………………………………….. 126
4.21 The spatially averaged v-velocity along the longitudinal direction is plotted
with the fluctuation, v', deviated from it at the location of ∆4, Re=5,500.
Solid symbol denotes the spatial average and open symbol denotes the
fluctuation……………………………………………………………… 128
xii 4.22 The temperature fluctuation, T' (K), is plotted together with the spatially
averaged temperature, Tavg as indicated in the figure; data are taken at ∆4.129
4.23 The component of Kd in the longitudinal coordinate varies along the height
of channel with respect to various flow rate (Reynolds number)……… 130
4.24 The component of Kd in the transverse coordinate varies along the height of
channel with respect to various flow rate (Reynolds number)………… 131
4.25 The thermal dispersion conductivity in the x and y direction versus the
Reynolds number under case (b)………………………………………. 132
4.26 The spatially averaged v-velocity along the longitudinal direction is plotted
with the fluctuation, v', deviated from it at the location of ∆4, Re=550... 133
4.27 Profiles of non-dimensional diffusivity in the transverse direction with
respect to various Péclet number, under wall condition of Case (a) –
non-symmetric heating………………………………………………….. 134
4.28 Profiles of non-dimensional diffusivity in the transverse direction with
respect to various Re, under wall condition of Case (b) – symmetric
heating…………………………………………………………………… 135
4.29 Plots of analyses under heating condition of case (a) to summarize thermal
dispersion study………………………………………………………… 137
4.30 Comparison of heat transfer effect in Nusselt number with respect to two
heating cases, case (a) and case (b)…………………………………... 138
4.31 Schematic of a three-dimensional porous duct with heated wall…….. 140
4.32 Contours of temperature and density of the porous system with meshes.. 141
4.33 Contours of density of Y-Z planes in the longitudinal direction of x… 142
xiii 4.34 The axial velocity contours at crossplanes (Y-Z) along the longitudinal
direction………………………………………………………………. 143
4.35 Detailed contours of axial velocity and temperature at the (a) mid-plane (∆3)
and (b)downstream (∆4 )of the porous sample……………………….. 144
4.36 Temperature contours in Y-Z planes along the axial direction x…….. 145
4.37 Contours of v, the velocity in the transverse direction of y at five Y-Z planes
along the axial direction x…………………………………………….. 147
4.38 Contours of w, the velocity in the transverse direction of y at five Y-Z
planes along the axial direction ……………………………………… 148
4.39 Velocity contours of three components in X, Y and Z directions at where x*
is of 0.25……………………………………………………………… 149
4.40 Four rakes of streamtraces are released at the upstream of the porous sample
at x* = 0.45 with contours of axial velocity. Each rake consists of 20
particles ocate at the heights of y* = 0.05, 0.25, 0.5 and 0.75 respectively
………………………………………………………………………... 149
B.1. 2D finite volume cells…………………………………………………... 164
xiv Acknowledgements
I am grateful to all those who have helped and encouraged me during my long journey in pursuing a Ph. D. degree. I am indebted to all of you. Your zeal for excellence and your personal warmth and kindness will continue to inspire me as I move forward to my next goals.
I must thank my advisor, Professor Isaac Greber for his guidance. I have learned a lot from him in both technical and non-technical discussions. I thank Drs. J. S. Tien, E.
B. White, R. V. Edwards and Mr. R. A. Blech for serving on my thesis committee, and for several helpful suggestions.
I wish to thank my husband, Meng-Sing and my boys, Yu-Ming and Deng-
Yuan. Without your support, I would not be able to go this far.
xv Nomenclature
A Channel cross-sectional area
Axi (Ax, Ay, Az), Cartesian components of the unit normal vector of
the cell face
B non-dimensional pressure drop based on the hydraulic diameter
Cf Inertial coefficient
Cv constant pressure specific heat for solid phase
c speed of sound
th ci molar concentration of the i species
D thermal diffusivity (K/(ρ·Cp))
D* dimensionless diffusivity (D/Din)
Dk numerical dissipation term
E vector of inviscid fluxes in the x-direction for gas-phase equations
2 2 2 Et total specific energy of gas phase, e + (u +v +w )/2
Ev vector of viscous fluxes in x-direction for gas-phase equations
F vector of inviscid fluxes in y-direction for gas-phase equations
Fv vector of viscous fluxes in y-direction for gas-phase equations
f non-dimensional effective heat conductivity of fluid phase in
porous system, [Kf]eff/ Kf
G vector of inviscid fluxes in z-direction for gas-phase equation
Gv vector of viscous fluxes in z-direction for gas-phase equations
h specific enthalpy of gas mixture
2 2 2 Ht total specific enthalpy of gas mixture, h + (u +v +w )/2
xvi Κ heat conductivity
K0 stagnant thermal conductivity of fluid phase
Keff effective thermal conductivity of fluid phase
Kd dispersion thermal conductivity
Kn Knudsen number, λ ⁄ l l local length scale of pore
L length of the numerical wind tunnel h height of porous samples ( height of channel as well) hsf interfacial heat transfer coefficient
n total number of cells
Nx number of mesh point in x-direction of the porous region
Ny number of mesh point in y-direction of the porous region
Ns number of gaseous chemical species
Nu Nusselt number (Equation 4.1)
p mean static pressure
Pe Péclet number (Pr·Re)
Pr Prandtl number (µCp/D)
qj heat flux in j-direction
qn net normal heat flux at the interface of solid and fluid cells
Re Reynolds number based on the hydraulic diameter (ρuh/µ)
Red Reynolds number based on the pore scale (ρul/µ)
0.5 Reκ Reynolds number based on the permeability (ρuκ / µ)
S source vector for gas-phase equation
xvii St Strouhal number
Tavg temperature averaged along the axial direction
T' fluctuation of temperature from the averaged temperature
T the temperature at the interface of solid and fluid cells
Tin inflow temperature
Th heated temperature at wall
Tm mean or volume averaged temperature (∫(ρuT )dA/ (ρVA))
* T invariant (dimensionless) temperature (Equation 4.2)
* U non-dimensional velocity magnitude with respect to uin
uin inflow velocity
V mean x-direction velocity = ((∫u dA/ A)
V' non-dimensional velocity vavg velocity in the y-direction averaged along the axial direction
(u',v', w') non-dimensional fluctuation components of velocity from the
respective averaged velocity components
th Vi volume taken by the i cell
Greek symbols
ρ mean density of gas mixture
µ viscosity of gas mixture
ε porosity
κ permeability (Equation 3.1)
τij viscous stress tensor component of Newtonian fluid
xviii λ mean free path
λmax the maximum eigenvalue of the preconditioning matrix
θ solid-to-fluid conductivity ratio ( Ks/Kf)
ω vorticity
∆n normal distance from cell center to face center
∆τ pseudo-time step
Subscripts f fluid s solid d dispersive eff effective w wall or surface quantity in inflow condition
Superscript
* dimensionless
xix A Numerical Study of Transport Phenomena in Porous Media
Abstract
by
May-Fun Liou
Since Darcy’s pioneering experimental study of porous medium flow, a great number of
analytical, numerical, and experimental works have been carried out to provide
qualitatively and quantitatively macroscopic descriptions of the overall viscous resistance
or heat transfer across the porous media. Recent advances in experimental measuring
techniques have uncovered the importance of structural heterogeneity within the porous
media. Thus, in order to gain a better understanding of phenomena at the scale of pores, new numerical approaches must be taken.
A general numerical simulation capability at pore-scale level is developed and validated in this thesis study, predicting global phenomena in close agreement with classical results. This technique has been successfully applied to two and three dimensional porous systems. In particular, it is shown that three dimensional solutions that couples the fluid and solid systems simultaneously at the pore scale are feasible with today’s computer resources and are extremely beneficial, shedding a new light into phenomena unavailable otherwise.
xx
This study also emphasizes numerical simulations of mass, momentum, and heat transfer
phenomena induced in complex porous media, providing details of local velocity profiles
and heat transfer. It is shown that the porous structures – shape, size, and locations have
significant effects on the macroscopic description. It is concluded that a microscopic description at the pore scale should be included in the study of porous medium flow. The flow pathlines are tortuous, determined by the local pore structure. Hence, the mixing caused by a porous insert can offer an efficient way to dissipate the heat from the heat source. The qualitative description of transport phenomena of flow through a three- dimensional duct demonstrates the capability of the numerical approach proposed in this thesis.
It is also found that the interplay among viscosity, heat conductivity and convection gives rise to a complex dynamical system. The effects of Reynolds number, Péclet number, local effective heat conductivity and properties of porous material on the local and global description of pressure, velocity field and heat transfer are studied in detail.
Finally, a summary of the thesis work and recommendation for future work are given.
xxi Chapter 1
Introduction
Natural and manufactured porous materials have broad applications in engineering
processes, including flow straighteners, heat sinks, mechanical energy absorbers,
catalytic reactors, heat exchangers, pneumatic silencer, high breaking capacity fuses, and
cores of nuclear reactors. For the subject of flow and heat transfer through porous media,
there have been extensive investigations covering broad ranges of applications since the early work of Darcy in the nineteenth century. Darcy correlated the pressure drop and flow velocity experimentally by defining a special constant property of the medium called permeability. However, it is only applicable to low speed (creeping) flow and low porosity saturated medium. It is well known that in flow through porous media the pressure drop caused by the frictional drag is proportional to the velocity at the low
Reynolds number range. In addition, this famous Darcy’s law also neglects the effects of solid boundary and the inertial forces on fluid flow and heat transfer.
Fluid transport is usually modeled using the continuum approach in terms of appropriate averaged parameters in which the real pore structure and the associated length scales are neglected. Moreover, those averaged parameters can only be obtained by experiments and are strongly influenced by the types of microstructure and operating conditions.
Fundamentally, they are limited to the scope of macroscopic phenomena. Specifically, the microscopic (pore scale) dispersion effect has significant impacts on the mass,
1 momentum, and thermal transports. Hence, modeling transport behavior at the pore-scale
for real engineering processes is desirable. In this study, an alternative numerical
approach is proposed and used for microscopic transport in porous media.
1.1 General Overview
Mass and thermal transport in porous media, such as ceramics, rocks, soils and catalytic
channels in fuel cells, play an important role in many engineering and geological
processes. There are two interesting aspects that arise in the research of porous media.
They are hydrodynamic and thermal effects. The dynamics of fluids flow through a
porous medium is a relatively old topic. Since the nineteenth century, Darcy’s law has
traditionally been used to obtain quantitative information on flow in porous medium. This
law is reliable when the representative Reynolds number is low whereas the viscous and
pressure forces are dominant. As the Reynolds number increases, deviation from Darcy’s
law grows due to the contribution of inertial terms to the momentum balance [Bear, 1972;
Kaviany, 1991]. It is shown that for all investigated media, the axial pressure drop is represented by the sum of two terms, one being linear in the velocity (viscous contribution) and the other being quadratic in velocity (inertial contributions). The inertial contribution is known as Forchheimer’s modification of the Darcy’s law
[Reynolds, 1900]. Basically, the pressure drop occurring in a porous medium is composed of two terms. Later Beavers and Sparrow [1969] proposed a similar model for
2 fibrous porous media. A general expression can be obtained from Bear [1972] and is widely accepted in the following formula,
dp µu =− (1.1) dx κ
It is seen that the pressure drop is directly proportional to the fluid viscosity µ and inversely proportional to the permeability of the porous mediumκ . Lage et al. [1997] suggested that an additional cubic term of fluid velocity be included in the above equation in the regime of higher speed (Reκ ≈ О(10)). Another significant work for predicting momentum transport in porous media is by Brinkman [1947]. Brinkman first introduced a term which superimposed the bulk and boundary effects together for flows with bounding walls. In Brinkman’s model, an effective viscosity was postulated from experiments performed on beds of spheres to replace the viscosity of fluid by taking into account of the porosity effect,
µeff = µ[1 +2.5(1 −ε)] (1.2) where ε is the porosity. There have been modifications on the above function to describe different types of porous media [Lundgren, 1972; Sahraoui, 1992]. More recently, computational modeling has been used to provide detailed flow fields. There are also results obtained by the asymptotic solutions [Chapman & Higdon, 1992].
While study of porous media flow is an old topic in fluid mechanics, the convective heat transfer of flows through porous medium has emerged as a new interest due to new technologies developments. Forced convection in porous media arises wherever the energy is delivered, controlled, utilized, converted or produced. The recent widely used
3 cellular microstructure materials have found implementation in the technologies of thermal dissipation media, impact absorbers and compact heat exchangers. Their thermal attributes enable applications as heat dissipation media and as recuperation elements.
Consequently, these enable high heat transfer rates and can be effectively used for either cooling or efficient heat exchange. Hence, it has become important to understand the interaction between mass and thermal transports and the resulting effects on the thermomechanical characteristics of porous media.
There has been considerable analytical, numerical, and experimental work done in the past to measure or estimate the overall heat transfer rate of convective heat transfer in porous media. Some early experimental studies have investigated the flow of fluids through packs of spheres [Lage et al., 1997; Kocecioglu & Jiang, 1994] or real porous material samples [Boomsma & Poulikakos, 2002]. Also, resurging interests in applying complex micro-structures to chemical industries, transpiration cooling etc., have prompted studies in experiments of highly porous media [Hunt & Tien, 1988a]. In general, the experimental results usually are described (postulated) statistically as empirical relationships in terms of dimensionless parameters of permeability, Reynolds,
Nusselt, Prandtl, and Peclet numbers. However, those correlations fail to address general mass and thermal transfer phenomena other than specific configurations, such as packed bed of particles, or highly porous open-cellular foams.
Most recently, non-intrusive experimental techniques have been applied to allow some detailed visualizations of microscopic flow and mass transfer occurring within porous
4 media. Among these are Laser Doppler Velocimetry (LDV), Particle Image Velocimetry
(PIV), and photoluminescent volumetric imaging (PVI). Dybbs and Edwards used a laser
anemometer for measurement inside model porous media. The measurement of local
velocity fields above the porous medium was first attempted by Vignes-Adler et al.
[1987], using LDV and Stephenson and Steward [1986] using PIV. Since then the experimental measurement techniques and devices have been advanced from differential pressure transducer and hotwires in packed bed of glass beads or cylinders [Beavers et al., 1974; Vafai, Alkire & Tien, 1985] to LDV and 100µm thermocouples [Okuyama &
Abe, 2000], or Magnetic Resonance Imaging (MRI) [Shattuck et al., 1991, 1995; Li et al.,
1994; Manz et al., 1999]. Notably, Saleh, Thovert and Adler [1993] used PIV to measure velocity close to porous media and pointed out the deficiency of the classical semi- empirical Brinkman equation [Brinkman, 1947]. However, these optical techniques can only be applied to transparent material and solid materials with optical index in the same range as the fluids. Meanwhile, Magnetic Resonance Imaging has been adopted for nonmagnetic materials to record the interstitial velocity distributions [Kutsovsky, et al.,
1996; Ogawa et al., 2001; Suekane, et al., 2003] for Darcian flow and non-Darcian flow respectively in packed beads and crushed glass samples. The obtained results for Darcian flow indicated a strong non-uniformity in velocity. In places, reversed flow is induced by pore structure. In particular, Sederman et al. [1998] observed a significant heterogeneity in the flow with 8% pores carrying 40% of the flow volume. Also, they found that high- volume flow rate is influenced mostly by the morphography of pores. These detailed measurements on the microscopic level indicate that mass and heat transfer in a packed bed of beads or crushed glass are sensitive to the interstitial velocity distributions in pores
5 over a wide range of Reynolds numbers (1 ≤ Red ≥ 210). These studies all point out that the structural heterogeneity of the porous region plays a critical role in the transport of mass and heat inside porous media.
Concerning analytical studies, Darcy’s law ignores the effects of solid boundary or the inertial forces on fluid flow and heat transfer. While these effects become significant near the boundary and in highly porous materials, relatively little attention had been directed to study these effects. Notably, Brinkman [1947] proposed a model to account for the presence of solid boundary by adding a viscous term to the Darcy’s law. Muskat [1946] took inertial effect into account by adding a velocity squared term to the Darcy’s law.
However, both works do not consider boundary and inertia effects simultaneously. Vafai and Tien [1981] used a volume-averaged momentum and energy equations to numerically study both effects simultaneously.
The volume averaging method currently is widely used for investigating inertial and boundary effects on flow and heat transfer in porous media. It uses local volume- averaging technique [Slattery, 1967 & 1970] and supplementary empirical relations to establish the macroscopic governing equations of momentum and energy for porous media [Vafai & Tien, 1981; Vafai et al., 1985; Calmidi & Mahajan, 1999]. These empirical parameters include the inertial coefficient Cf in the momentum equation and the
effective thermal conductivity Keff in the energy equation, which consists of stagnant K0 and dispersion thermal conductivity Kd. Conventionally, the stagnant thermal conductivity is determined from porosity and the thermal conductivity of the solid and
6 the fluid phase. Moreover, some models consider extra terms of interfacial heat transfer to account for the diffusion between two phases [Wakao, 1979]. An empirical model for its determination was presented by Tien and Vafai [1979]. In more advanced approaches, a geometric averaging technique is used for estimating heat transfer [Hsu, 1994 & 1995].
Also, Calmidi & Mahajan [1999] and Boomsma & Poulikakos [2001] developed models by utilizing geometrical estimate for calculating the effective thermal conductivity, specifically for metallic foams saturated with fluid. Calmidi & Mahajan proposed a one dimensional heat conduction model of a two dimensional array of hexagonal cells representing the porous medium. Later Bhattacharya et al. [2002] extended the Calmidi &
Mahajan model with a circular intersection, which results in a six fold rotational symmetry. Also Boomsma and Poulikakos used tetrakaidecahedron cells with cubic nodes at the intersection of two nodes to model highly porous metal foams. Both geometrical models are based on experimental data to provide the geometric parameters.
Recent applications of Lattice-Boltzmann methods provide pore-scale information and give promising results [Martys & Hagedorn, 2002; Manz & Gladden, 1999; Verger &
Ladd, 1999; Szymczak & Ladd, 2004]. The major difficulties in those approaches are how to model flows in heterogeneous materials and how to properly handle the interfacial boundary conditions between fluid and solid matrix at the pore scale [Bauer, 1993;
Thompson & Fogler; 1997; Thompson, 2002].
Though the local volume-averaging method is a more practical approach than the analytical counterpart, it does not provide a general approach in describing transport in
7 porous media. Since the volume averaging is still in use today and is the framework for
studies of porous media, the next section will be devoted to the volume-averaging
method.
1.2 Volume Averaging Methods
y+∆y
up y
x v+(∂v/∂y)∆y x+∆x z+∆z y+∆y
z w+(∂w/∂z) ∆z
u+(∂u/∂x) ∆x u
w
y x Representative x+∆x Elementary v Volume (REV)
Figure 1.1: Schematic of porous medium, flow variations in the Representative
Elementary Volume and the pore velocity vector u p .
The local averaging performs the averaging of the microscopic equations over a
representative elementary volume (REV) as shown in Figure 1.1. Note here, the REV is
8 the smallest volume possessing local averaged quantities that are statically meaningful.
Its definition and the relevant mathematical operations for partial differential equation
(e.g. N-S equations) may be found in Slattery [1967] and Kaviany [1991]. A scalar
quantity φ of a fluid is averaged over a fluid volume Vf ,
1 φφ =dv (1.3) V ∫ vf
Let the porosity of a porous medium ε = Vf / V, then
1 ⎛⎞1 φφ = dv = ε ⎜φdv⎟ (1.4) VV∫ ⎜⎟∫ vvff⎝⎠
1 ∇φφ=+∇ φ dA (1.5) V ∫ f Asf
and
1 ∇ iiΦΦ =+∇ Φ idA (1.6) V ∫ Asf
whereΦ may be a vector or a tensor.
After applying Eqs. (1.3)-(1.6), the microscopic governing equations of continuity,
momentum and energy are averaged and the resulting equations become their
macroscopic counterparts. Reviews by Kaviany [1991] and Calmidi [1998] are good
sources for detailed derivations and discussion. The final forms are given below (with the volume averaging symbol, ~, dropped).
∇iu = 0 (1.7)
9 ∂ uuu µ 2 ρ[()+ ∇=i ]−∇ p +∇u+r (1.8) ∂t εε ε
()ρCTpf∇iiu =∇{(Keff+Kd)∇T} (1.9) where r in Eq. (1.8) is the total drag force per unit volume exerted by the solid phase and can be expressed as
1 µ r =− ∫pdA+ ∫()∇uidA (1.10) VVAAfs fs
It can only be evaluated if the geometry of the porous structure is known exactly. But these kinds of details are exactly what the averaging approach attempts to avoid initially.
Hence, by requiring to know details at the pore level, it would contradict the concept and advantages of the volume averaging method. Thus, an empirical model of r is adopted instead. A commonly used model incorporates the Darcy-Forchheimer model, Eq. (1.2), with the parameter κ and Cf determined by experimental means. It is also noted that u and
T refer to volume averaged velocity and temperature in Equation (1.9). Equation (1.9) is a single temperature treatment which assumes the solid phase and fluid phase being in thermal equilibrium state. Therefore, the heat transfer caused by molecular conduction is lumped into a bulk property which is called the effective conductivity, Keff. This assumption is only valid in the porous system of similar thermal properties among its constituent phases. The thermal dispersion conductivity, Kd, accounts for the thermal dispersion effect caused by the interaction of fluid and solid matrix. Similarly, both Keff and Kd can only be evaluated if the geometry of the porous structure is known exactly, like r in the momentum equation. For a more general approach, coupling two energy
10 (temperature) equations for the solid and fluid phases is needed. As in the volume averaging momentum equation, Equation (1.8), semi-empirical models are proposed
[Hwang, 1994; Wakao, 1979; Nield & Bejan, 1999].
The resulting equations from inserting these described models into Equations (1.7)-(1.9) are summarized here as,
∇iu = 0 (1.11)
∂u 1 2 εµ C f ρε ρε( +∇iuu) =−∇p +µ∇u + u + ||u ||u (1.12) ∂t εκκ
∂Tf u ερ()CT[+∇ii]=∇{[(K)+K]∇T}+ha(T−T) (1.13) pf ∂t ε f feff d f sf sf s f
∂T (1 −=ερ)( C) s ∇i(K ∇T) −ha(T−T) (1.14) ps∂t eff s sf sf s f
where || u||= uiu. The parameters (Kf)eff and (Ks)eff refer to the effective thermal
conductivities of respective phases, namely fluid denoted by “f” and solid by “s”; hs f and
aTs fs(−Tf) are used to achieve the coupling between the two energy equations of solid and fluid phases. It also represents the energy transfer between two phases due to temperature difference. For the limiting case of local thermal equilibrium between the solid and fluid phases, the resulting simplified energy equation of the porous system is,
()ρCTpf∇ii(u )=∇[(Kfe)ff+Kd]∇T (1.15)
11 To close the above system of governing equations, Eqs. (1.11)-(1.14), the following quantities need to be provided from experiments, which should be conducted with the porous material of interest. It’s obvious that the details of the transport occurring at the pore-scale and their effects on macroscopic level are lost in the averaging process. To make up these pore-scale transports, the following empirical constants are introduced in the modeling of transport phenomena associated with all scales. These coefficients are,
(1) The effective heat conductivities of solid and fluid, ( K eff )s and ( Keff )f,
(2) The permeability κ and inertial coefficient, C f ,
(3) The dispersion coefficient Kd ,
(4) The interfacial heat transfer coefficient hs f and the interface area as f .
In reality, flow inside porous medium is not necessarily in the ideal states of being fully developed, having uniform porosity, being local thermal equilibrium or isotropic medium. Hence, it is desirable to capture the transport phenomena that occur at the length scale of the pores. To study these interesting local pore-scale phenomena, for example, shear stress on solid boundary, mixing occurring around solid etc., and their effects on the macroscopic transport is the objective of this thesis. To achieve these goals, a different approach is proposed in this study.
However, a full solution by solving simultaneously the Navier-Stokes equations for fluid and heat transfer equation for solid at the microscopic level has not been obtained before for two and three dimensional structures in the open literature. Thus, in this thesis, a
12 novel numerical approach is described to address the multi-scale modeling challenges. A general numerical simulation capability at pore-scale level is developed and validated.
Meanwhile, this study will emphasize numerical simulation of mass and heat transfer phenomena induced in complex porous media in terms of local velocity profiles and heat transfer.
1.3 Governing Equations
This section describes the full Navier-Stokes equations for fluid phase and the heat transfer equation for solid phase, together with their constituent thermo-physical properties of both phases. These are the governing equations used for the thesis study, without any further simplifications.
1.3.1 Fluid-Phase Equations
The equations representing the gas-phase flow are the three-dimensional, unsteady, density weighted and time-averaged forms of the compressible Navier-Stokes equations.
Gravity force is neglected. In Cartesian coordinates, these equations at the microscopic level are given by:
∂∂Q ∂ ∂ +−()EE+(F-F)+(G-G)=S (1.16) ∂∂tx vv∂y ∂z v
13 Q = (ρρ, uv, ρ, ρwH, ρ , ρY, ρY,..., ρY )T tN12 s −1 E =+(ρρuu, 2 p, ρuv, ρuw, ρuH, ρuY, ρuY,..., ρuY )T tN12 s −1 F =+(ρρρv, v, v2 p, ρvw, ρvH , ρvY , ρvY ,..., ρvY )T tN12 s −1 G =+(ρρρwww, , v, ρw2 p, ρwH, ρwY, ρwY,..., ρwY )T tN12 s −1 E = (0,ττ, ,τ,uτ++vwττ+q,ξ,ξ,...,ξ)T (1.17) v xx xy xz xx xy xz x x12x x Ns −1 F =+(0,ττ, ,τ,uvτ τ+wτ+q,ξ,ξ,...,ξ )T v yx yy yz yx yy yz y y12y y Ns −1 G =+(0,ττ, ,τ,uvτ τ+wτ+q,ξ,ξ,...,ξ )T v zx zy zz zx zy zz z z12z z Ns −1 S = 0 where the superscript “T” denotes the transpose of a vector. The variables ρ, P, u, v, w,
Ht, K, and Yi represent the density, gauge pressure, x-, y-, z- cartesian velocity components, total enthalpy, and species mass-fraction. The vector Q contains the conservative variables, the vectors E, F, and G are the inviscid fluxes, and the corresponding viscous fluxes are denoted by the subscript “v”. The source term S vanishes because no radiation, body forces, chemical reactions, or phase change is
22 2 considered. The total specific enthalpy is Ht = hu++(v+w)/2 and the total specific
energy EHtt=−p/ ρ . The notation for the rest of variables is rather standard and their meanings are listed in nomenclature.
It is noted that in this research the fluid phase is represented by ideal gas of air. Hence, phenomena of importance in small scale liquid flow, such as surface tension, capillary force and adsorption are not present in this study.
14 1.3.2 Solid-Phase Equation
Solid regions within the flow domain are described by the unsteady heat conduction equation for an isotropic material with a specific internal energy satisfying desolid =
СυdТsolid. If radiative heat transfer is considered, it can be treated via a suitable boundary condition at the solid–fluid interface.
∂T ρ Cs =∇i(K ∇T) . (1.18) s v∂t ss
1.4 Selected Materials for Study
In the present study, the fluid studied is limited to air and the solid is selected from various types of material. Their thermo-physical properties, taken from Engineering Tool
Box [2005] and used in this study are listed in the Table 1. Solid phase is assumed isotropic with constant properties (in reality they are varying, e.g., with temperature but very slightly in the problems studied). Contrary to the solid phase, the thermo-physical properties of fluid phase vary and are determined by the states of the flow. This study covers solid phase materials from metal to rubber. Aluminum is lightweight, strong, corrosion resistance and has good heat conductivity. It has broad applications in manufacturing and transportation sectors. Silicon Nitride is used as in high temperature structure ceramic due to its superior heat resistance, strength and hardness. Applications include: heat exchangers, nozzle, engine components and valves. The third material chosen is Silicon rubber. It is commonly applied to extreme temperature resistance in
15 automobile and aviation. The last one is a hypothetical material which exhibits properties same as that of air. It is devised for the purpose to study dispersion only which is caused by channeling effect occurring inside of porous medium. Therefore, a group of diverse two-phase systems is selected, covering a wide range of thermal conductivity ratios of their constituent phases (θ = Κs / Κf) from 8320 to 1.
3 Solid material Κs (W/K-m) Cp (kJ/(kg-K)) ρ (kg/m ) θ
Aluminum 218 0.91 2,700 8320
Si3N4 25 0.5 2,300 954
Silicon Rubber 0.166 2.0 1,250 6.34
Air(hypothetical solid) 0.0262 1.005 1.016 1
Table 1: Thermo-physical properties of various solid phases used in the present study [Price,2001; MatWeb, 2005].
1.5 Outline of the Thesis
This thesis is organized as follows. Chapter 2 details an alternative computational approach to simulate flow through porous media at pore-scale. The basic and necessary ingredients are described and discussed.
Chapter 3 is the study on pressure drop and provides detailed descriptions of mass and momentum transports occurring in porous medium. Computational tests to validate this
16 method for relevant problems have been carried out. Then, computations for a configuration simulating a porous flow experiment in a wind-tunnel. The technique introduced in Chapter 2 is applied to quantitatively predict pressure drops at various porosity and topography of porous medium. Comparison with known experimental and theoretical data available in the open literature for porous flows is conducted. Numerical results are gathered and discussed in terms of permeability, κ, and the dimensionless pressure drop P*.
Chapter 4 is concerned with forced convection occurring in porous media with various thermophysical properties. Two numerical settings are considered. One is a two- dimensional channel flow and a three-dimensional duct flow at various inflow and heating conditions. Four kinds of porous media are investigated, see Table 1. The major focuses are on the thermal dispersion and heat conduction in porous media. Results are presented using some relevant parameters, for example, effective conductivity and heat transfer coefficients.
Finally, Chapter 5 summarizes the thesis study and suggests future studies.
17 Chapter 2
An Enabling Technique for Pore-scale Simulation
2.1 Introduction
Although the modeling of fluid transport in porous media has been extensively studied in the past, the length scales involved and the complexity of pore structure make such studies mathematically and numerically challenging even up to the present day. The detailed geometry of the pore structures, aside from statistical (or manufacturing) variations, can be given through microtomography or microscopy [Spanne, 1994; Coles,
1998]. Such studies are limited to only small representative samples, giving information for only small portions of perhaps a more complicated porous structure. Even if one knows the detailed structure, a general exact solution describing the mass and heat transport in these complex pore geometries is difficult to obtain. Most historical and current reported work can be categorized as being either (1) theoretical simplified or approximate, (2) experimental, (3) numerical or (4) combinations of the above. The present study adopts a numerical approach.
In the numerical modeling approach, most models employed so far treat the porous medium as a single phase continuum, using the volume averaging approximations which were discussed in the preceding chapter. For these models, a priori knowledge of spatially averaged transport parameters [Vafai & Tien, 1981; Vafai et al, 1985; Calmidi
18 & Mahajan, 1999] is required. Often, those parameters can only be obtained through experiments. As a result of the cost and time required for experiments, these models are not generally applicable for materials over a wide range of pore composition, shapes, and concentrations. In addition, there is a real concern as to whether the averaged parameters at the macroscopic level can really be representative of multiple scales of fluid transport in this type of problem. If the length scale of interest is small by comparison with the scale over which the parameter is averaged (such as occurs in the dispersion effect), then averaged parameters are not suitable for describing any phenomena of smaller scale.
Some other researchers have developed numerical models that involve directly solving the momentum equations in the microstructure of the porous medium. However, these models involve a variety of simplifying assumptions. For example, a bank of parallel cylinders or a bed of parallel spheres in lattices are often used to represent fibrous or granular materials, respectively. Assumptions of lattice periodicity and material homogeneity are common in these models. Some limiting cases of this sort can be solved analytically. For example, Larson and Higon (1987) studied Stokes flow in a periodic array of cylinders. In these parallel lattice models, fluid flows around a regularly spaced array of separated solid objects, instead of flowing through tortuous interstitial gaps in a solid matrix. Thus, the phenomena of mixing and of recirculation of local fluid (which are classified as secondary flow effects), and the resulting thermal dispersion [Calmidi &
Mahajan, 2000], are neglected. In addition, the exclusion of wall effects, variable porosity, and heterogeneous materials in this kind of approach limit the applicability of analyses to only special cases. A recently developed network model [Thompson 1996,
2002], and the emerging Lattice Boltzmann model, have been applied to simulating
19 porous medium transport, and their merits in this regard have been demonstrated [Martys,
2002; Verger & Ladd, 1999; Szymczak & Ladd, 2004]. However, in Lattice Boltzmann models, the lack of precise knowledge of the location of the pore/solid interface is a potential source of error. In addition, the handling of interfaces between solid and fluid in Lattice Boltzmann model can be tedious and computing-intensive due to its rigid lattice structure.
In general, these aforementioned numerical efforts involve one or more of the following assumptions, which limit the applicability of the approach to special situations:
(1) inertial effects are negligible (Darcy’s flow),
(2) the boundary/viscous effect (Brinkman friction) is insignificant,
(3) local thermal equilibrium (LTE) applies between fluid and solid,
(4) dispersion effects are negligible,
(5) the flow is laminar,
(6) the thermo-physical properties of the fluid are constant,
(7) the flow is thermally fully developed, and
(8) the solid matrix is of constant porosity (i.e., it is macroscopically homogeneous).
In the present work, an alternative numerical modeling approach for flow simulation in porous media is proposed and studied. In this approach, the multi-phase (solid and fluid) and multi-scale problem is numerically simulated by solving simultaneously the three- dimensional Navier-Stokes equations in the fluid-filled pores and the heat transfer equation in the solid matrix, and this is done over the entire spatial extent of the porous
20 medium. The approach is designed to directly compute the microstructure-related pore- scale transport phenomena within accurately represented geometric microstructure, and for a wide spectrum of microstructures. Because of this, there is need neither for predetermined or empirical macroscopic averaged transport properties, nor for assumptions of periodicity or homogeneity of the microstructure. Further, the approach minimizes the use of simplifying approximations in the modeling of both the geometric microstructure and the transport phenomena. The principal assumptions regarding the transport phenomena are that the fluid and solid matter are each a continuum, i.e., that molecular effects are adequately described by the usual material point thermodynamic and transport properties, and that these continua are isotropic. The first assumption is valid only when the Knudsen number of fluid is small, and the solid matter granules or fibers are much larger than the molecular size.
The rest of this chapter describes the essential elements and methodology of this simulation approach. Section 2.2 describes the technique of geometric discretization of the porous medium. Then, examples of its applications in two-dimensional porous media and three-dimensional ones are provided in Sections 2.3 and 2.4, respectively. Section
2.5 describes the transport equations and their numerical solution method. Section 2.6 describes test cases and validations, and is followed by a discussion in 2.7 and a chapter summary in 2.8.
21 2.2 Mesh-Based Microstructure Representation Algorithm
Some previous porous medium transport simulation models [Martys, 2002; Verger &
Ladd, 1999; Szymczak & Ladd, 2004] have accounted for the microstructure by incorporating both the solid phase and the fluid phase as distinct phases in the geometry of the domain, but they are generally accompanied by simplifying approximations in the transport equations. Typically, a simplified set of transport equations is solved in the fluid regions, possibly together with the heat conduction equation in the solid regions. In the conventional approach, a prescribed physical porous domain of interest is mapped to a computational domain, sometimes by periodic extension of an optical scan of a microtomograph of a representative porous sample, followed by a labeling of solid and fluid regions. This is in itself a difficult task, especially when attempting to numerically represent a three-dimensional porous microstructure. Next, the computational domain is geometrically discretized by mesh generation, and finally the governing equations are discretized on the mesh and solved for unknowns at each grid point or cell center by using an appropriate numerical algorithm, subject to corresponding boundary conditions.
In the second step, of mesh generation, major difficulties arise from the need to respect the geometry of the fluid-solid interface in order to accurately capture the pore-scale transport phenomena. If a structured multiblock mesh is generated, the difficulty of gridding the numerous tortuous pore passages is extreme. Even if an unstructured mesh is generated, the need for the mesh to conform closely to a very complex boundary structure places great demands on the mesh generator. Furthermore, the meshes on the fluid and solid sides of an interface must conform to each other at the interface, or else computationally expensive interpolation will need to be carried out at the interface, often
22 with a resultant loss of accuracy. These difficulties in generating a quality mesh that respects a physically prescribed convoluted fluid-solid interface, have in the past made a full Navier-Stokes solution of flow in porous media seem infeasible. Therefore, typically either the microstructure geometry that has been vastly altered so that it can be described by a regular periodic lattice, or only a global parameter such as porosity is employed, which masks the dynamics within the porous region.
In the present study, the difficulties outlined above are circumvented by a new approach to the computational domain definition and mesh generation, which reflects the random nature of the spatial distribution of the pores. In the conventional approach, a prescribed physical domain with existing fluid-solid interface is mapped to a computational domain, and the latter is geometrically discretized by mesh generation. In the new approach developed here, the reverse sequence of steps is followed. First, the outline of the computational domain is selected to represent the physical domain of interest. Next, the computational domain is discretized with a fine enough mesh, which is generally an easy task since there are no convoluted interfaces to conform to. Then mesh locations are chosen as seed locations of solid material; these locations are chosen randomly, but with prescribed rules. The solid regions are then allowed to grow, again randomly but in accordance with prescribed rules. The rules are established so that the eventual statistical properties of the porous region matches closely the geometric properties (such as the porosity and the pore size) of the physical porous medium that it is sought to simulate.
The solid and fluid-filled pore regions are then assigned the thermodynamic and transport properties of the corresponding physical materials. As a natural byproduct of this process, the interface between solid and pore regions always lies at the boundary between mesh
23 cells. Thus it results in a complete conforming at the solid and fluid interfaces. Because the mesh cells act as a bed or base from which the porous medium is constructed, this approach is termed a mesh-based microstructure representation algorithm. This algorithm has three advantages over the conventional approach in that: (1) the mesh generation is an easy task, (2) the meshes automatically conform to the solid-fluid interface and to each other at the interface, and (3) no laborious transfer of the solid-fluid interface geometry from the physical to the computational domain is required.
This methodology entails three steps, namely generating the mesh, selecting seeds, and growing the solid matrix from seeds; they are described in the following sections, and illustrated schematically in Figure 2.1. A mesh generator is used to generate a bed of unstructured or structured meshes in the computational domain. This mesh bed is used both for generating the solid matrix and for performing the numerical computations. The mesh used herein is triangular and quadrilateral for two-dimensional and tetrahedral and hexahedral for three dimensional geometries. Seeds, which are initial solid elements, are selected within the mesh bed. The seed locations are chosen using a random number generator which has been well validated for its uniformity of randomness [Matsumoto &
Kurita, 1994]. For more details of the random number generator, see Section 2.2.2. These seed cells are grown spatially into a geometry determined by imposed constraints; for example a fiber-like or sphere-like solid matrix as used in the current computations, The processes of seeding and growing are iterated until the desired pore quality and porosity are achieved. In summary, the procedure is:
(1) Generate a mesh
(2) Put in locations of seeds of solid within the mesh
24 (3) Grow the solid around the seeds, in accordance with some controlling rules
(4) If a particular porosity is desired, repeat (2) and (3) until the desired porosity is
achieved
(5) Possibly refine mesh for the purpose of resolution
(6) Find behavior of fluid flow and corresponding temperature field, and perhaps
other properties, by using a numerical solver for governing equations.
In Figure 2.1, controlling inputs for each procedure are listed within the procedure box while the resulting schematic output is shown underneath. The arrow signs of ↑↓ indicate iteration between two processes. Options for pore shape include sphere or fiber, but are not limited to these two. Orientation is one of geometric inputs determining the tortuous character of fibrous materials.
25 Generate mesh structured/unstructured/hybrid
In: Number of cells
Structured/unstructured quad mesh bed Unstructured triangular mesh bed
Call Random number generator to seed solid matrix
In: Number of seeds
26
Growing pores from seeds
Input : Pore shape, distribution, orientation, & Min/Max distance between neighboring pores/solid block
27 Iterations Stop at desired porosity
Figure 2.1: A schematic explaining the procedures of generating a numerical porous sample.
This methodology is developed by utilizing the existing software available in grid generator and numerical solver and algorithms to enable numerical studies of porous medium at pore scale. The factors of randomness and prescribed controlling are taken into account and included in the discussion of the following sections.
2.2.1 Mesh Generator
Surface modeling and grid generation technology have played a critical role in CFD analyses. Their developments are mainly addressed to support CFD analyses and to bridge the connection between the CAD (Computer Aided Design) tool (definition of solid geometries) and calculations of flowfield about the geometry. The grid generation has been generally categorized into either a structured or an unstructured approach. They both have advantages and shortcomings and an user’s decision of choosing one over the other is largely based on the problem to be studied, ease of dealing with geometry, accuracy of interest. It is widely accepted that the structured grid can more easily provide higher accuracy solution than the unstructured grid, but is more difficult to generate for a complicated topology such as multiply-connected bodies. It would present a nearly impossible task to employ a CFD solver based on the structured grid approach to deal
28 with many vastly irregular pore regions in a porous medium. Hence, it is concluded that a
CFD solver based on the unstructured grid approach1 is the only feasible avenue for possibly tackling the geometry issue, which is the foremost one in any numerical discretization and is especially so in the present study. Based on their geometric differences, this study uses quadrilateral grids (brick-like in 3D) for fibrous solid matrix and triangular grids for sphere-like porous structure. It is noted that the unstructured quadrilateral mesh is a better choice than structured grids because it gives more flexibility in controlling the shape and size of solid matrix and in managing the data structure.
To give a perspective view of grid generation with regard to the geometrical requirements in this study, the following attempts to briefly describe the prevailing grid generation methods that are used most often today.
.
(1) Block-structured mesh
The standard procedure is to first generate surface grids on block faces, which can be either a physical geometry boundary or interior computational block interface, from point distributions placed on the face edges by distribution functions. Then volume grids are generated within the block, either via algebraic transfinite interpolation functions or via a set of elliptical partial differential equations. The former can easily provide orthogonal grid lines (surfaces), but with loss of generality. The latter, on the other hand, is more general, but is hard to maintain orthogonality within the block or at the block interfaces.
The structured grid generation has been researched and developed intensively in the past
1 It is remarked that an unstructured mesh can utilize various types of polygons or polyhedrons, including quadrilateral and hexahedron.
29 four decades. Today, there are several publicly or commercially available codes, such as
EAGLE (Thompson, 1987ab), GRIDGEN, ICEM and NGP (National Grid Project). In general, each code provides the following functions: automatically determining the connectivity between geometric and grid entities and writing out adjacency and orientation information when features of graphically interactive and automatic modules are invoked.
(2) Unstructured mesh
There are two most common ways of generating unstructured grids, namely, the
Advancing Front Technique (AFT) [Lőhner, 1988 a & b] and the Delaunay approach
[Weatherill, 1992; Mitty et al., 1991]. The AFT introduces one element at a time, while
Delaunay introduces a new point at a time. Both methods are a scalar approach but differ in numbers of operations required to introduce a new element or point.
Advancing Front builds the tetrahedral cells progressively outward from the triangulated body surfaces, with element size controlled by some distribution function, often interpolated from background grid. Special procedures are required to prevent the overlapping of fronts which are advancing toward each other. In addition, smoothing and repair may also be necessary at places where fronts meet.
In the Delaunay approach, grid generation starts from a coarse triangulation created only from boundary points. It is followed with an insertion of points according to the specified resolution. Then the triangulation is locally reconstructed according to the Delaunay
30 criterion that the circumsphere through the four vertices of a tetrahedral cell does not contain any other grid point. The first good Delaunay refinement algorithm is due to
Chew [1989]. To address the need in CFD analyses, especially inside a boundary layer, high aspect ratio triangles growing outward from the boundary are desired. Often, some of these high aspect ratio meshes are too skewed to provide numerical accuracy.
However, isosceles with aspect ratio of unity is not practical in this region if computation cost is of concern. For example, a hundred isosceles are needed to replace a triangle with aspect ratio of 10. Well-shaped tetrahedral can be obtained with algorithm proposed by
Shewchuk [1997]. Other commonly used techniques are combinations of AFT and
Delaunay triangulations [Muller, 1994; Mavriplis, 1993].
In general, block-structured grids require fewer words of storage per grid point and can take advantage of factored and directional solvers. Tetrahedral grids work well for Euler solution but not for Navier-Stokes solvers. Hybrid grids combine the best features of structured and unstructured grids. In practical applications of CFD analyses, it may not be desired to have a uniform mesh of unity aspect ratio, due to several factors. These may be related to physical phenomena in which the flow gradient in one direction is much higher than the other, e.g., in boundary layer, or related to geometrical reality, e.g., in fibrosis. Hence, non-uniform triangles or tetrahedra may be of necessity. In fact, grid generation systems in all production codes are developed for capability in handling grids of large variations for providing flexibility in encompassing large aspect ratio cells with good numerical properties.
31 (3) CFD-GEOM grid generator
The methodology proposed in this study is so general that it can work with various mesh generators. In this study, a commercial software program named CFD-GEOM©, developed by CFDRC is used to provide the meshes needed. CFD-GEOM© CFD-GEOM of CFD Research Corporation [Hufford et al., 1995] is an interactive geometric modeling and grid generation system for block-structured grids, tetrahedral (advancing front) grids and hybrid grids. The geometry is NURBS (Non Uniform Rational B Splines) -based, reads IGES (Initial Graphics Exchange Specification) files and has some internal CAD capabilities. The system has macro library capability. For unstructured grids, GEOM provides user tools to locally refine mesh distributions by adding source points or lines.
Users can also impose constrains on cell size and curvature [CFDRC, 2002]. Two classes of mesh: structured and unstructured and their combination (hybrid) are applied in this study; the mesh sensitivity issues with respect to pore distribution and pore density are examined and presented in Section 2.6.
2.2.2 Random Number Generator
A good random number generator has to satisfy statistical properties that are frequently obtainable only by numerical experiments. A pseudo-random number generator with properties of a long period, a low serial correlation, and a high order of equi-distribution, is desirable. Statistical tests are performed with numerical simulations. Generally, a pseudo-random number generator is used to estimate some quantity for which the theory of probability provides an exact answer. Comparison with this exact answer provides a measure of how well the generator simulates "randomness". There are many software
32 choices that are free for download from various Internet websites or are commercially available. A review of pseudo-random number generators can be found in [L'Ecuyer,
1998]. The current mesh-based algorithm is not restricted to any specific random number generator.
In this study, Mersenne Twister (MT) is adopted to generate an array of uniformly distributed random numbers, either as integers or double precision floating point numbers, for use in the seeding process. MT is a pseudo-random number generator developed by Makoto Matsumoto and Takuji Nishimura [Matsumoto & Nishimura,
1998]. It is designed with consideration of the flaws of various prior generators. It has a far longer period and far higher order of equidistribution than most other generators, with an assured 2 19937-1 period and a 623-dimensional equidistribution property. Although the calculation speed depends on the computing system, it is claimed that MT can be faster than the standard ANSI-C library random-number routine in a system with pipeline and cache memory. The program is written in all three of FORTRAN 90, C++ and C, and is listed in Appendix A.
In those cases that non-uniform pore distributions are desired, a non-uniform distributions random number generator is required. In the simplest cases a non-uniform distribution can be obtained analytically from the uniform distribution of a random number generator by applying an appropriate transformation. More complicated distributions can be created by the acceptance-rejection method, which compares the desired distribution against a distribution which is similar and known analytically. This usually requires several samples from the generator. Another commonly used approach uses various cumulative
33 functions and their inverse functions to determine the upper and lower tail of the desired distribution. Several options are used, for example, the Gaussian distribution or
Laplacian distribution [Press et al., 1986]. Figure 2.2 shows a simple non-uniform porosity sample.
Figure 2.2: A porous sample with non-uniform porosity.
2.2.3 Quality Control
Control of quality in the growing process is important because the approach proposed above is mesh-based, rather than based on the precisely specified porous microstructure.
The solids and pores are made out of groups of neighboring cells with each cell in a given solid or pore region carrying the same tag of solid or fluid, respectively. To ensure the numerical stability of the solution algorithm and accuracy of the solution, one need be concerned with the issue of the number (or physical size) of computational cells required
34 in order to fairly represent an individual granule or pore in the physical domain. From the pure accuracy point of view, one wishes the finest mesh possible. But the resources are limited. It is not feasible to do so in practice because the finer the mesh is, the more computational time is required, which generally increases much faster than the number of mesh elements because there are more elements to evaluate and there will be more time steps taken to achieve convergence. Hence it is a decision based on acceptable accuracy, practitioners’ previous experiences of validation exercise, similar problems, etc.
Moreover, the mesh size required for a given accuracy is certainly problem-dependent, varying with flow parameters such as Reynolds number, Péclet number, geometry complexity, flow phenomena (complexity), etc. That this issue concerning mesh size/number is arguably still an open question is evident from a recent survey paper
[Baker, 2005] which reports the finding from an international workshop on drag prediction. For lack of a definite quantifiable guidance, the present study adopts to use meshes as sufficiently fine as possible to resolve micro-scale phenomenon but within a manageable limit of computing cost.
Several control factors are estimated and iterated to guarantee the quality of numerical porous medium. These include pore shape, pore size, minimum distance among neighboring solids, total number of meshes, and topology of mesh used. Several parameters can be used to control these procedures. If seeding location, shape and size of each solid matrix are considered and minimum distance is mandated, there are eight scenarios. In this thesis, only two scenarios are studied.
35 (1) Random seeding location but size and shape predefined
The seeding location can be random in terms of physical location or numerical location.
Adopting physical location as a criterion gives a direct sense of location, thus easier to understand. However, these two criteria are equivalent, only differing by a mapping function which is inherent in mesh generation. Hence, the choice comes down to computational convenience. Here, randomness in numerical location is chosen over physical location because the latter does not always coincide with cell center. The elimination of heavier distribution in denser grids region is already controlled by the criteria of minimum grid resolution between neighboring solid matrices. In practical cases, for example, ceramics used in fuel cell applications and filter design in purifying applications, pores are manufactured with a preferred distribution instead of being complete random.
(2) Random seeding location, shape and size
This combination adds more numerical complexity. The only controllable parameter is the prescribed margin which defines the minimum number/distance of grids among solid matrices. A converged result from procedures described in methodology is more difficult to obtain.
A filter is installed to ensure no cavity is formed inside of solid matrix. Although this last condition was applied for all the calculations in the present study, it is not a limitation of the present methodology, which can also quite easily deal with the case of isolated cavities being formed inside the solid matrix during the manufacture of the porous medium.
36 2.2.4 Porosity
Porosity is defined as the ratio of the volume of the total pore space filled by the fluid to the total volume of the porous medium, all within the domain of interest. Since mesh cells in the domain have been tagged as solid or fluid, the volume of total pore space can easily be obtained numerically by finding the sum of volumes of the cells tagged as fluid.
Similarly, the volume of the solid matrix is the sum of volumes of the cells tagged as solid. Finally, the total medium volume is simply the sum of the volumes of all cells, see also Figure 2.3. Then the porosity is defined by the following equation
nfluid ntotal ε = ∑Vi∑Vi (2.1) ii==11
V s
Fluid
Solid
Figure 2.3: A Schematic of typical domain of a porous medium – two phase problem.
For some materials which have non-constant porosity [Benenati & Brosilow], there are various techniques available to generate numerical porous media with the same non- uniform characteristics. In addition to those techniques that are described in Section 2.2.2
37 for pre-defined random number distributions, other applicable techniques can also be implemented at various levels of the media generation process. For example, some possible techniques are: (a) usage of locally clustered meshes for the fluid phase, (b) usage of mass sources in the process of generating meshes, and (c) specification of pore radius as function of physical location.
2.3 Formation of a Two-Dimensional Geometry
Figures 2.4(a), (b), and (c) are three representative porous media obtained numerically.
These three plots are used to demonstrate that various solid microstructures can be obtained from triangular, rectangular, and hybrid meshes respectively. They exhibit different characteristics determined by inputs provided in the processes of seeding and growing inside structured mesh beds. The blue color represents fluid elements and red for solid in these samples. The sample (a) is fibrous, of low porosity, and of no fixed orientation. The sample (b) is sphere-like, small radius (d ≈ 10-5 m), and of high porosity.
Lastly, sample (c) is formed in a hybrid mesh and with larger sphere shape. Other examples of porous media are presented in following chapter. Therefore, they are not shown here.
38
(a)
(b) (c) Figure 2.4: Randomly formed solid matrix representing porous medium in 2-D domain (a) fibrous shape formed with triangular mesh (b) small solids in circular shape formed with triangular mesh, (c) large solids in circular shape with hybrid mesh.
39
2.4 Formation of the three-dimensional geometry
The recent advances in the manufacturing technologies of open-cell foams have broadened their engineering applications. Those applications include enhancing heat transfer in adsorption heat pumps [Guilleminot and Gurgel, 1990], rocket liquid propellant combustion chambers [Fottini and Tuffias, 1998], and high power electronic devices [Antohe et al., 1996; Bastawros, 1998; Lu et al., 1998]. Open-cellular foams can be characterized by a consolidated solid microstructure composed of a series of interconnected ligaments capable of continuous thermal conduction. An open-cellular material with a prescribed porosity of ε can easily be numerically generated. First, a three-dimensional mesh bed of tetrahedral, prism, quadrilateral or hybrid mesh is used in the same manner as the two-dimensional one described in Section 2.3. The procedures are similar to the two-dimensional case, and they are described in 2.2, including seeding, growing, quality control and convergence to generate a sample of porosity of 1−ε and with a sphere-like solid structure in the cross-section, which constitutes a closed-cell foam. Then the cells which had been tagged as being fluid are retagged as being solid, and vice-versa. This toggling of the tags yields a highly porous open-cellular material with the prescribed porosity.
Figure 2.5 shows the microstructure of an industrial grade open-cellular ceramic foam. In
Figure 2.6 a simple geometry is used to schematically demonstrate the process of numerically generating similar material.
40
Figure 2.5: Open-cell foam microstructure.
Figure 2.6 (a): The cross-section of a typical mesh-based porous medium is shown with solid balls in black and space taken by fluid in white.
41
Figure 2.6(b): The cross-section of numerical generated open-cellular sample after converting all solid and fluid tags from Figure 2.6(a), into the oppose ones.
The mesh used herein is represented with triangular/rectangular cells. A sample porous medium obtained numerically is presented in Figure 2.6(c).
Figure 2.6(c): 3D Numerical fluid and solid mesh in cross cut planes along the axial direction – x direction.
42 2.5 Numerical Solutions
With the discretized geometry established, appropriate computer codes, for example,
Navier-Stokes equation solvers, or molecular dynamics codes may all be used to produce the level of numerical accuracy and efficiency as appropriate. An existing Navier-Stokes code is adopted to incorporate the methodology described in the above sections. The code is currently referred to as National Combustion Code (NCC). Though NCC is a set of tools for the analysis and design of combustion systems, it can also be used for non- combusting flowfield calculations [Moder et al., 2000]. Another reason for choosing the
NCC is that it contains a generalized thermodynamic and transport properties database for fluid flow composed of arbitrary number of gaseous and liquid-phase species. Only those features of the NCC that are relevant to this study will be briefly described in the following sections.
In this work, the three dimensional Navier-Stokes equations of fluid flow, Equation 2.2, are solved, in conjunction with the heat conduction equation of solid matter, Equation
2.3. Note here that the solid matrix is assumed to be isotropic and homogeneous. That is, the solid matrix has the same mechanical and thermal-physical properties everywhere, and they are independent of orientation.
2.5.1 Governing Equations
The set of equations appropriate under the concept of this thesis study will entail a simultaneous solution of both fluid and solid phases. The equations representing the
43 fluid-phase flow are the three-dimensional, unsteady form of the compressible Navier-
Stokes equations, as given in Chapter 1 and Appendix B. Solid regions within the flow domain are described by the unsteady heat conduction equation, also given in Chapter 1 and Appendix B.
2.5.2 Boundary Conditions
At the solid and fluid interface, the velocity of fluid has to vanish, which is the non-slip condition for continuum flow of a viscous fluid. The temperature and the heat flux have to match at the boundary of solid and fluid elements.
u = 0, f w
T= T (2.2) f wws ,
ddTT K= K. f( )(f s)s dniidn
th where, ni is the normal direction at the surface of the i element.
Since the physical interface between solid and fluid is aligned with the numerical boundary of mesh cells, which are cell faces for a 3D geometry or cell edges in 2D, the interface conditions can be easily enforced as is done in a body-fitted mesh. The matching conditions at the fluid-solid interfaces are zero fluid velocity and continuity of
condition ״no-slip״ temperature and heat transfer rate, Equation (2.2). This is the regular applied to both velocity and thermal fields. A detailed description is given in the following.
44 Once the desired numerical porous matrix is obtained, each cell in the whole computational domain is tagged as either solid or fluid. The interface between two phases can be easily located by checking all edges (2D) or faces (3D). At an interface which has one cell tagged as fluid cell and the other connecting cell as solid, shown in the following,
Solid phase Fluid phase
* n , qn *
∆ns ∆nf
the normal distance from each cell center (denoted by *) to the edge/face center (shown as a dashed line) is computed as ∆n of solid and fluid and the edge/face normal (→) points from the solid phase cell to fluid phase cell. Then the temperature at edge/face center T is defined by the continuity of normal heat flux at the face.
Knfs∆ TTs + f Knsf∆ T = (2.3) Kn∆ 1+ fs Knsf∆
and the net heat flux at face is
K f qn=± (T −Tf) (2.4) ∆n f where the sign “±” is determined by the definition of normal direction. The heat flux is always pointed towards the fluid phase in this study, hence “-” is taken. It is noted that
45 both the temperature and heat flux at the solid and fluid interface given above now can be conveniently employed in the discretized energy equation. Also, the continuity of these two quantities is automatically enforced. It can be shown easily that the second-order accuracy of temperature is preserved and the heat flux at interface is obtained through the first-order differencing.
There is no need for additional terms to represent the drag force exerted by the pore walls on the fluid, or for inclusion of bulk porosity or effective thermal-physical properties, both are required in the traditional volume averaging approach. It is quite easy to incorporate surface reaction mechanisms between a reactive fluid and active solid material, for instance in the active catalyst layer and electrode in a fuel cell simulation.
2.5.3 Numerical Methods
The gas-phase transport equations are spatially discretized using a central differencing scheme. The blended Jameson type 2nd and 4th order dissipation terms are added to avoid numerical instability and odd-even decoupling. For steady-state solutions, the gas-phase equations are marched in pseudo-time by an explicit four-stage Runge-Kutta scheme.
Local pseudo-time and residual smoothing are used at each stage to increase the time-step under the stability condition, thereby accelerating the convergence to steady state. The same numerical scheme is applied to solid-phase equation with a relaxation factor which reduces the oscillations in solution. For detailed explanation on numerical methods
46 mentioned above, Roache [1972], LeVeque [1990], Anderson, Tannehill & Pletcher
[1984] have excellent descriptions, and a brief description is given in Appendix B.
2.6 Sensitivity Test
2.6.1 Geometric Types
Though there are many distinct types of porous media, they can be classified into two major kinds if only geometrical shape is considered. The two kinds are granular/closed- cell, and open-celled. There are granular porous materials and closed-cell foams, for instance sand, crushed rock underground, etc. In general, these types of material can more or less be treated as packs of balls or cylinders and are low on porosity. The other kind is the open-cell structure, fibrous type materials which conventionally are made out of metal or carbon and exhibit high porosity. They are also commercially available in copper, aluminum, nickel, and the non-metallic substance of polyurethane. Since the true structure is considerably complex, an exact representation of materials is not achieved here. In this study, the solid structure is approximated by two types of geometric representations, sphere-like and fiber-like ones.
2.6.2 Grid Independence Study
It is expected that the more the mesh is refined, the more closely the micro-scale can be simulated. In the vicinity of the solid-fluid interface, there exist large velocity and temperature gradients, and an adequate number of mesh points is necessary to accurately
47 resolve these variations. Also, the use of fine grids in the flow direction may be critical when the effects of axial conduction and convection are of concern. However, the desire for finer meshes is limited by available computing resources. Hence, it was necessary to conduct a grid-independence test before this mesh-based methodology could be used for further computations. The test was done by varying mesh sizes while holding porosity to a constant. Table 1 summarizes the results obtained in this grid-dependence study. The studied case is a 2D channel flow with porous sample located in the middle of the channel. The flow is fully developed before it reaches the porous section. The inflow, outflow and boundary conditions are kept the same throughout this numerical sensitivity test. A structured mesh was used, with fibrous solid matrix, porosity of 0.83, and
Reynolds number Re of 1093. The first column in Table 2 refers to the spacing in the x and y directions. The second and third columns refer to the total number of grid points in the x and y directions, respectively. The fourth column is the dimensionless pressure drop across the porous medium. The case number is used for discussion.
6 (∆x, ∆yw) m Nx Ny B*10 Case #
(4*10-4,2.5*10-4) 51 201 2.31 1
(2*10-4,2.5*10-4) 101 201 3.04 2
(4*10-4, 1.9*10-4) 51 251 2.33 3
(2*10-4,1.9*10-4) 101 251 3.05 4
(1.3*10-4,1.9*10-4) 151 251 3.057 5
Table 2: Five cases are computed with various combinations of mesh sizes in axial and transverse directions for grid dependence study.
48 Grid points are clustered along the y direction in the regions near the bounding walls.
The ∆y used here is the local length scale of the first grid point away from wall. Both ∆x and ∆y are in meters. The dimensionless pressure drop, B, is used to measure the required mesh resolution in both x and y directions. It can be concluded that ∆y of 10-4 m is fine enough in the y-direction, i.e., the cross flow direction, if Case 1 and Case 3 are compared. For these two cases, ∆y varying from 2.5*10-4 to 1.9*10-4 has negligible effect on the computed pressure drop. Since initial pore locations (seeds) are chosen by a random number generator, two sets of initial seeding lead to almost same amount of pressure drop. It shows that flow is insensitive to where seeds are planted so long as the porosity and the shape of solid matrices are kept the same. Hence, ∆y of 2.5*10-4 m, which has been shown to be adequate for obtaining a converged solution, will be used for all computations to minimize the required computing resources. Similarly, the ∆x of
2.0*10-4 m is chosen for the same reasoning.
2.6.3 Convergence Study in Terms of Residual
Another important check of the validity of a computation is an examination of the history of convergence. Figure 2.7 represents a typical computation, Case 1 in the preceding section, in which convergence histories obtained from solving both Navier-Stokes equations for fluid and heat transfer equation for solid structure are displayed.
The decimal logarithms of the residuals of fluid and solid equations are plotted against the iteration number. There are 50000 structured cells used in this case. The solutions are
49 determined to be converged only if both solid and fluid solutions converged. The figure showed that both solutions converged at about the same rate, with the convergence of fluid solution slowing down in the end.
Figure 2.7 : Convergence history of residuals from solving Navier-Stokes equations for fluid and solving heat conduction equations for solid in the case of the channel flow heated from both top and bottom walls.
2.7 Discussion
This new modeling technique for porous medium transport not only allows for better quantification of how microscopic properties affect macroscopic transport, but also has been demonstrated to be quite robust. The geometrical shape of the solid phase is determined by the type of mesh of which it is composed. The pore density is currently controlled by random number generator; it can be extended to include defined geometries. The geometry of the numerical pores/granules generated in this study would
50 not be a perfect circle or sphere, and would possibly be sharp-edged ligaments in open- cell form or exhibit a bit of zigzagged fibrous shape. Nevertheless, such characteristics mimic those of the real manufactured porous media which come with slightly varying pore size, and sometimes even with rough edges [Lu, Stone and Ashby, 1998].
2.8 Summary
The technique introduced here is general and easy to implement in an existing CFD code.
The quality control of pore sizes and pore distributions are critical to its success in providing an alternative for pore-scale simulations. However, care must also be exercised while checking the mesh quality, as this also controls accuracy in all numerical simulations. There are limitations to the technique, and some remedies to overcome these limitations. To produce reliable solutions at the pore-scale, adequate amount of grid points across the pores are required. In low porosity porous media, the void volume occupied by fluid is much less than in the high porosity ones. Hence, with a prescribed uniform mesh spacing, this is equivalent to fewer cells and less resolution available for solving the Navier-Stokes equations. To meet this requirement of sufficient resolution, the technique of adaptive meshing can be incorporated to locally refine meshes further in those regions that require it. The second limitation of the current technique is that the flow is no longer continuum in nature if the local characteristic length (finest mesh scale) is comparable to the mean free path of the fluid (Knudsen number >>1). Though the behavior of the fluid flow near the wall in not yet fully understood, replacement of the non-slip condition with a Maxwellian slip condition at the interface or wall [Arkilic et al.,
51 1997] is a common practice in the range of 10-3 ≤ Kn ≤ 0.1 [Barber & Emerson, 2002;
Löfdahl, 1999]. In conclusion, this technique is robust in accurately predicting transport in porous media while accounting for pore-scale phenomena. It is computationally more expensive than conventional simplified approaches, but has a great potential with the continuously evolving cheaper and faster computing resources.
52 Chapter 3
Fluid Flow and Pressure Drop
3.1 Introduction
The objective of this chapter is to demonstrate that the modeling technique described in
Chapter 2 is capable of numerically providing correct the pressure drop incurred in a flow through a porous medium.
For convenience of the discussion in this Chapter, a widely accepted description of pressure drop, quadratic in velocity, is given explicitly:
2 dp µu Cuf ρ =− − (3.1) dx κ κ where the first term is the well known Darcy’s term as described earlier, the second term
is a quadratic dependence on velocity and C f is called the inertia coefficient.
Previous studies have shown that the permeability κ and the inertia coefficient Cf vary with the structure of solid matrix. The values of κ and Cf have been obtained in several ways, e.g., by using an analytical capillary model, packed balls, hydraulic radius model
(note that these are for media of low porosity) in periodic cells or straightforward experimental measurements. Beavers [1969] made the earliest attempt to quantify the
53 pressure drop for high porosity medium. Equation 3.1 has been applied to numerous numerical studies of porous media.
Table 3 is a summary of what have been accomplished by the past and current research efforts. The following sections include discussions of these subjects: validation tests, numerical setup for studying the pressure drop with respect to pore structure, and numerical results reporting the fluid’s hydrodynamic behaviors in porous media.
Formulation and Assumption References Method Major results Periodic porous Larson & Higdon, Solving Stokes flow excellent accuracy, media, low 1992 with a periodic grain moderate Reynolds number consolidation computational flow model, collocation effort method used Incompressible, Verberg & Ladd, solving Stokes flow, study of the low Reynolds 1999 Lattice-Boltzmann convergence of the number flow method used permeability as a function of grid resolution for random arrays of spheres - a second order approach. Periodic porous Chapman & solving unsteady a study in the media, low Higdon, 1992 Stokes equations, dynamic Reynolds number oscillatory pressure permeability and flow gradient imposed acoustic propagation in porous medium Incompressible, Martys & Hagedorn, Solving Brinkman Evaluating low Reynolds 2002 equation for Stokes permeability in number flow flow, Lattice- multiple pore size Boltzmann method material - low used porosity, using parallel computing technique Incompressible, Sangani & Yao, Stokes flow Longitudinal Low Reynolds 1988 equation permeability a
54 number, weak function of Suspended random porosity if the arrays circular arrays porosity ≤ cylinders 0.7; transverse permeability also insensitive of porosity if porosity ≤ 0.5.Solving actual flow gives better estimation of permeability than solving disturbance of flow. Incompressible Fand et al., 1987 Experiment providing a simple method to characterizing the behavior of porous media in the transition region between Darcy and Forchheimer and between Forchheimer and turbulent flow. Compressible, high Masha et al., 1974 Experiment Incompressible Reynolds number Forchheimer resistance law still valid for subsonic flow with significant density changes.
Table 3: Summary of some works in evaluating viscous resistance (pressure drop) in porous media.
55 3.2 Validation Cases
As test cases of the mesh-based methodology two laminar cases involve fluid phase interacting with a stationary single solid body were considered, a square and a circular cylinder.
3.2.1 Laminar Flow over a Square Cylinder
The first case is a creeping flow (v = 0.0001 m/s) in which the fluid flows over a square cylinder situated in a boundary free domain with ReD of 0.5 (based on the height of solid body). The configuration and grid points, consisting of 14400 elements, are presented in
Figure 3.1. The fluid flows from left to right. The adiabatic and no-slip conditions are imposed on the surface of cylinder. Periodic conditions are applied at the top and bottom surfaces of the computational domain. After the Navier-Stokes equations are numerically solved, the streamlines can be obtained to demonstrate the nature of top-bottom symmetry about the axis of symmetry of the body in the flowfield. This symmetrical result comes from the linearity of the governing equations, a characteristic of low
Reynolds number (say, O(10-100)) flow.
56
Figure 3.1: Streamlines of a low Reynolds number flow around the solid block shown in blue color, exhibiting symmetry between the top and bottom half planes.
3.2.2 Laminar Flow over a Circular Cylinder
Next, a two dimensional flow over a circular cylinder is considered to test the ability of the NCC code for capturing an unsteady flow around a single solid body at a higher
Reynolds number. This is a standard benchmark case for unsteady flow problem, and various experimental and computational references are available in the open-literature, e.g., Roshko [1953] and Visbal [1985].
A cylinder of diameter of 0.1 m is considered. A hybrid mesh is used with an O-type of mesh surrounding the cylinder to help facilitate solution convergence and reduce numerical oscillations. A total of 6603 elements are used in the computation, see Figure
3.2. The insert at the right upper corner of the Figure 3.2 shows the mesh zoomed around
57 -4 the solid cylinder. The first grid spacing away from the cylinder, ∆yw, is 5x10 m. The adiabatic and no-slip conditions are imposed on the surface of cylinder. The left half of the outermost grids is assigned the inflow conditions (u = 0.1 m/s, v = 0 m/s, ρ = 0.187 kg/m3) while the other half is imposed with a fixed pressure p and with extrapolations of other variables from their interior values. The resulting ReD (based on diameter) is 64, lying in the range in which the existence of unsteady vortex shedding is ensured
[Fornberg, 1980].
Figure 3.2: A hybrid mesh over the circular cylinder, consisting of an O-type mesh around the cylinder – enlarged view given in the inset. The contours of longitudinal velocity component u are plotted in Figure 3.3. The contours indicate that the flow is fully developed in the flow field. A small reversed flow, denoted
58 by the dark blue color, is formed behind the cylinder. The computed Strouhal number (St) is 0.161, a close agreement with the experimental values of 0.16-0.17, reported by
Roshko [1953].
u 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01
Figure 3.3: Velocity contours showing vortex-shedding behind a circular cylinder (ReD=64).
Considering the above two single solid body cases, the first one demonstrates qualitatively how the numerical code performs in the regime of Darcy’s flow while the second case provides a quantitative comparison with experimental data. Therefore, computations with capabilities of dealing with solid bodies of arbitrary numbers and sizes in the porous media can be pursued with confidence. In subsequent sections and chapters, various flow problems of interest will be discussed in detail.
59 3.3 Numerical Experiments of 2D Porous Flow
The schematic of the configuration is shown in Figure 3.4. The direction of air flow is from left to right and the porous sample is located in the middle of the domain (20 cm x
10 cm) to allow for a well-developed flow before it reaches the porous sample and in order to minimize any numerical instability. The distance from inlet to the front side of the porous medium, is 9 cm, which is also about the range estimated by Kaviany [1991].
Five representative axial positions, ∆1, ∆2, ∆3, ∆4 and ∆5, as indicated in Figure 3.4, are selected for evaluating data and interrogation of flowfield; they are located respectively upstream, starting plane, mid-plane, exit plane and downstream of the porous medium.
The porous sample length is 2 cm. The flow rate varies by changing the inflow velocity uin. The pressure drops are calculated based on the pressure difference across the porous sample.
Flow exits uin
ρ P Porous Medium Tin y x ∆ 1 ∆2 ∆3 ∆4 ∆5
Figure 3.4: Schematic of porous flow in a 2D channel and stations for flow analysis.
60 Several representative pore structures have been studied. Table 4 lists structures of different porosities with two types of geometric shapes.
Shape Case Porosity d (mm) No. of solid Total no. of elements Elements Structure 1 0.854 3 650 47,000.
Sphere- Structure 2 0.89 3 454 47,000. like Structure 3 0.947 3 235 47,000.
Fibrous Case 4 0.83 0.75 1705 50,000.
Table 4: Morphologic characteristics of porous samples used in the pressure drop study.
In the case of sphere-like porous medium, d is the radius of the sphere, while d is the thickness of fiber for fibrous solid phases. The fibrous structure is composed of slender and long fiber in sheets in which the form drag is much small than the circular cylinder tubes structure. The listed total number of elements (both fluid and solid) in Table 4 is the number of total cells used in the computation.
3.4 Results and Discussion
The numerical procedures are set to mimic the experiments, conducted either in wind- tunnel or water-tunnel. Porous samples are inserted inside a channel in two-dimension and a duct in three-dimension to simulate the experiments performed in the manner
61 described in Hunt & Tien [1988a & b], Calmidi [1998], and Okuyama & Abe [2000]. The numerical domains used are usually extended from the experimental one to ensure the flow is fully developed before it reaches the porous sample, in a similar manner to that in a wind/water tunnel in which the flow is smoothed through an extended section before the test section. This addition of an extended domain also helps minimize any numerical instability. In the following simulations, a variable spacing is used in both X and Y directions (longitudinal plane). Meshes close to the walls and in the porous section are finer than the rest of domain. The smallest mesh along the X and Y-directions is on the order of 10-5 m.
3.4.1 Two-Dimensional Channel
The physical dimension is 20 cm x 10 cm, in which 60,000 elements are used. The porous sample has a length of 2 cm and is situated in the middle of the channel, located between x=0.09 m and x=0.11 m. Its geometric structure of pores, the distribution, location, and scale are indicated in black in Figure 3.4. The porosity is 0.83. The computed results are sampled and presented at five locations along the X-direction as shown in Figure 3.4. The notation, ∆1, ∆2, ∆3, ∆4, and ∆5, correspond to locations at x=0.05 m, 0.09 m, 0.1 m, 0.11 m and 0.15 m respectively (porous region lies between ∆2 and ∆4). Several different porous structures are analyzed. The aim is to study how the structure of porous medium plays a role in the mass transport and momentum transport, especially the pressure drop. Numerical studies are performed in a range of Reynolds number, Red ~ 0.5 - 400. A typical inflow velocity is in the range of 0.1 m/s to 1.0 m/s.
62 At the inlet, the fluid enters with a specified mass flow rate and temperature, m and Tin respectively. Results and discussions are arranged in two parts, (I) and (II). Part (I) mainly focuses on the study of the mass transfer in the porous medium. Part (II) presents the relation between parameters of ReЄ , porosity (ε) and the dimensionless pressure drop.
3.4.1.1 Velocity field
The walls at top and bottom of the channel are isothermal, kept at the inflow temperature,
Tin, of 300 ˚K. The inflow velocity is at 0.1 m/s. The solid phase is made out of silicon nitride with porosity of 0.83. Its thermophysical properties are listed in Table 1, Chapter
1.
Figures 3.5 (a) and (b) give the distributions of non-dimensional velocity magnitude (U*) and velocity components (u' and v') in the axial (X) and vertical directions (Y) respectively. These variables are non-dimensionalized with respect to the inflow velocity uin. Since the flow is primarily in the axial direction, it is not surprising to see that the profiles of U* and u' are quite similar. At the exit of porous medium, the velocity profile colored in purple is no longer parabolic or as symmetric as the velocity profile upstream
(at ∆1 or x*=x/L=0.25) of porous medium in red color line. At ∆2 (x*=0.45), ∆3 (x*=0.5), and ∆4 (x*=0.55), fluid either accelerates or decelerates locally according to the converging or diverging path inside the porous medium. There are upward or downward deflection of flow, v', inside the porous medium, as shown in the Figure 3.5(b). A reverse
63 flow is seen in the region where the flow begins entering the porous medium, as shown in u' versus y' in Figure 3.5(b).
U*
Figure 3.5(a): The distribution of non-dimensional velocity magnitude along the distance from the bottom wall, y', with respect to sampling locations of ∆1, ∆2, ∆3, ∆4, ∆5, and at the exit of the channel.
64
Figure 3.5(b): Velocity distributions of u' and v' in the axial (x/L) and vertical direction (Y/H) respectively. The variables u' and v' are non-dimensional velocity, with respect to the inflow velocity and y' is normalized by the channel height.
65 The velocity contours are presented in Figure 3.6 (a). The dark blue zones indicate either the solid matrices or the nearly-stagnant fluid in the rest of flowfield. This above described distortion in velocity may cause concerns in some engineering applications.
Contours displayed in Figure 3.6(b) are the vorticity induced by the heterogeneous velocity distribution within the porous structure which in turn significantly promotes mixing of mass, velocity and temperature at pore scale. This is interpreted as thermal dispersion in the literature and this subject will be examined extensively in Chapter
4.
Figure 3.6(a): Contours of velocity magnitude (Evec) in the entire numerical domain for the inflow velocity of 0.1 m/s; the area colored in blue indicates zero velocity and where solid phases are located
66
Figure 3.6(b): Vorticity contours caused by fluctuating velocity within porous structure, showing the vorticity strength much higher inside the porous region than the rest.
Figures 3.7(a) and (b) are plots of local pressure distributions versus the longitudinal coordinate x at several y locations. They show that the major pressure drop occurs within the porous medium. Those fluctuations in front and after the porous medium are due to the presence of the porous medium which produces disturbances that propagates in all directions since the flow is at very low speed. In addition, Figures 3.7(a) and (b) indicate that the local pressure distributions are determined by the geometric structure of the porous medium. This phenomenon agrees with experimental observations, in which the
MRI technique is used to detect correlations between porous structure and flow by
Sederman, et al. [1998] and Jones et al. [2000]. Note that the pressure plot indicates that the drop across the porous sample is significant, compared to the drop due to wall friction in the rest of channel. This is found to be true for the range of velocities computed, even for low Re cases, and it is also confirmed by experiment conducted by Calmidi [1998].
One observes that there are no significant differences in the x-pressure variation with y location.
67
Figure 3.7 (a): Pressure distributions along the axial direction with respect to several y locations close to the wall.
68 Figure 3.7 (b): Pressure distribution along the axial direction at three y locations, y’ = 0.25, 0.5 and 0.75. In (b) the pressure is described at the y-locations beyond that in (a).
A series of similar analyses were conducted for porous media with different geometrical characteristics, for example, porosity and geometric shape of solid phase. For these cases, the flowfields exhibit the same flow mechanism as the above representative case qualitatively.
69 3.4.1.2 Pressure Drop and Porosity
Figure 3.8 presents the effects of solid shape and permeability on the sphere-like and fibrous structures listed in Table 4.
ε 0.85 0.89 40 0.94 0.83 (fibrous)
30
P*
20
10
0 10 20 30 40 50 60
Re d
Figure 3.8: Non-dimensional pressure drop, P*, of various porous systems in Table 3 are plotted against the modified Reynolds number. Open symbols are for results from sphere like structure and the closed one is for fibrous structure.
2 The non-dimensional pressure gradient parameter is defined by P*= (∆p/L)l /µuin. Here
∆p/L is the overall pressure gradient and d is the characteristic solid size. Here ∆p/L (∆p
=p(∆4)-p(∆2) is the overall pressure drop, l is a characteristic pore size, µ is the
70 coefficient of viscosity, and uin is the approach velocity. Note, the non-dimensional pressure gradient parameter is equivalent to the dimensionless permeability, κ/l2. The variable P* is used instead in the following discussion. In Figure 3.8, it is evident that the sphere-like structure causes higher pressure drop than the fibrous structure.
P* ε=0.61 500
400
ε=0.83
300
200
100
0 100 200 300 400 Red
∗ Figure 3.9: Non-dimensional pressure drop P vs. Red in a wide range of Reynolds numbers for two values of porosities.
The pressure drops (P*) with respect to the flow conditions (Red) are shown in Figures
3.9 and 3.10 for two values of porosity. Figure 3.9 shows the behavior over a large range of Reynolds numbers in which the behavior in low Reynolds number range is shown in a
71 close-up view in Figure 3.10. One sees P* approaches a value near 10 as Reynolds tends to zero and there is a larger range where P varies linearly with Reynolds number; this means that the pressure gradient varies as the square of the approach velocity. Thus the
Darcy limit and Forchheimer flow are observed. Moreover, the morphological effect is insignificant if flow is in the low Reynolds number range.
35 P* ε=0.61 30 ε=0.83 25
20
15
10
5
0 d 0 5 10 15 20 Re
∗ Figure 3.10: A close-up view of the non-dimensional pressure drop P vs. Red, exhibiting different behaviors caused by varying porosityε .
72 3.5 Summary
The velocity variation within the porous medium depends on the structure of the porous media, as manifested via the macroscopic pressure drop in Figure 3.8. This behavior exhibits hydrodynamic mixing at the pore scale. Its importance will be re-addressed in the next chapter. The pressure drops occurring across the porous medium is attributed to several factors, including form drag, viscous drag from bounding wall and inertia force.
The obtained results confirm that the pressure drop is a linear function of flow velocity at low Reynolds number regime and a quadratic function at higher Reynolds numbers. The morphological effect is an additional factor in determining pressure drop. The pressure drop is an accumulated result of all factors described above.
73 Chapter 4
Forced Convective Heat Transfer in Porous Media
4.1 Introduction
As pointed out earlier, since the late twentieth century, with the growing concern of high cost of energy and fuel, technologies involving convective heat transfer in the flow of energy-carry fluids through porous media, for example porous insulation and the cores of nuclear reactors, have gained interests. Hence, knowing transport properties of fluid- saturated porous material are very important in wherever the energy is delivered, controlled, utilized or produced. Examples include petroleum, geothermal industries, fuel cells, etc. In addition, forced convection is a critical part in the cooling of electronic equipment. The topic has been dealt with in textbooks [Kaviany, 1991; Bejan, 1995], and general articles review on forced convection in porous media [Lauriat & Ghafir, 2000].
In the past, an important issue concerning numerical simulation of flows through porous media based on volume averaging techniques is the choice of model to be employed in the governing equations to perform the simulation. Most models are results from experimental or theoretical data for some specific porous systems and simplified conditions. Alazmi and Vafai [2000] have investigated in the open literature variants within the porous media transport models and classified them into four categories:
74 constant porosity, variable porosity, thermal dispersion, and local thermal non- equilibrium. In these works, major assumptions are made, such as: steady, incompressible, isotropic/homogeneous properties of the porous medium and fluid constant thermophysical properties of the fluid and porous medium, and inclusion of only transverse dispersion while neglecting longitudinal one. They have found that variances in these models within each category have a pronounced effect on the velocity field and a substantially smaller effect on the temperature field and Nusselt number distribution.
However, there are no universally superior models valid for a range of applications, such as thermal insulation, packed bed heat exchanger, catalytic reactors, geothermal systems and electronic cooling. As such, most efforts in modeling transport phenomena in porous media, including those being briefly reviewed later in this section, are limited to at lease one of the following assumptions:
(1) inertia effect negligible (Darcy’s flow),
(2) viscous term (Brinkman friction) insignificant,
(3) local thermal equilibrium (LTE) between fluid and solid,
(4) constant thermophysical constitutive properties of fluid phase,
(5) flow thermally fully developed,
(6) constant porosity/ homogeneous solid matrix,
(7) axial conduction term insignificant (Large Péclet number).
The assumptions (1) and (2) have been addressed in Chapter 3. This chapter will address the effects of the remaining assumptions and the pertinent works published by others on these subjects. For comparison purposes, five aspects will be analyzed and discussed in detail, which will distinguish the approach of this thesis from the widely-used volume-
75 averaged numerical approach in modeling transport processes through porous media.
Specifically, they are variable porosity (VP), variable (non-constant) thermophysical properties of fluid phases (VT), local thermal non-equilibrium (LTNE), thermally developing porous systems (TD) and thermal dispersion (TDP).
(I) Variable Thermophysical Properties of Fluid Phase (VT)
In general, the energy equation (Equation (1.16), Chapter 1) of fluid phase includes terms of conduction, convection, and heat source in all directions, in which the thermophysical properties may be functions of states and spatially varying. However, uniform properties of viscosity, specific heat capacity of fluid are commonly used. Ling and Dybbs [1992] presented a theoretical investigation of the effects of a temperature-dependent fluid viscosity on force convection for a flat plate flow. The results show a strong influence on heat transfer when the fluid viscosity was modeled as an inverse function of temperature.
Recently, some of the above simplifying assumptions have been relaxed by Narasimhan et al., [2001] and Nield et al. [1999]. They proposed a theory to predict the thermo- hydraulic effect on the fully-developed convection by a fluid with a temperature- dependent viscosity, flowing through a porous medium channel. In Narasimhan et al.
[2001a, 2001b], the numerical analyses of a parallel-plates channel with constant heat
o o flux at bounding plates are presented in the range of 5 ≤ T ≤ 170 . Nield et al. [2003 &
2004] extended their earlier study of temperature-dependent viscosity [1999] to include the effect of viscous dispersion, axial conduction in forced convection in a parallel-plate channel. The variance considered in the above-mentioned works is only in viscosity,
76 while variations in density, specific heat and thermal conductivity are neglected. In this thesis, all constitutive properties are determined by each species of the fluid phase and a chemistry package, Chemical Equilibrium with applications (CEA) [McBride & Gordon,
1993], which provides the required species thermodynamic, transport and kinetics data.
The mixture’s molecular viscosity and thermal conductivity are obtained by using
Wilke’s law and Mathur’s law [White, 1991] respectively. The species properties Cpi are given by a fourth-order polynomial of temperature.
(II) Local Thermal Non-Equilibrium (LTNE)
Traditionally, the assumption of local thermal equilibrium has been used in analysis of heat convection in porous media. That is, any temperature differences between the solid and fluid phases are neglected. Thus, the problem of flow in porous media can be simplified from a two phase to a single phase one. However, under certain situations, the local thermal equilibrium is not valid where a substantial temperature difference exists between solid and fluid phases. Thus, a two-medium treatment is necessary. Kaviany
[1991] proposed a heuristic two-temperature approach to deal with a condition when there is an internal heat source in one of the media, or if the thermal conductivities of the fluid and solid are disparate, such as the air versus metal foams. There have been studies on convection in packed beds and open cellular metal foams [Hunt, 1988, Sathe, 1990;
Sozen, 1993; Hwang, 1994; Amiri, 1994; Hwang, et al., 1995] to account for local thermal non-equilibrium. In this work, thermal non-equilibrium is taken in which the
77 three dimensional Navier-Stokes equations of fluid flow, Equation (2.2), are solved, in conjunction with the heat conduction equation of solid matter, Equation (2.3).
(III) Thermally Developing Flow (TD)
In a recent literature survey of over 30 papers by Nield and Bejan [1999], each of these papers is based on the assumption of negligible axial conduction, i.e., the Péclet number is sufficiently large. Therefore, the flow can be treated as thermally fully developed, thereby yielding simplification in the analysis. Hadim [1994], Nield et al. [2002] and
Poulikakos & Kazmierczak [1987] studied thermally developing convection within either low or high porosity porous medium. In this study, the two (parallel-plates channel) and three dimensional (duct) configurations are partially filled with porous media. Hence, the solution of the general governing equations employed here will naturally evolve, rather than the form being pre-chosen, to determine whether the flow is thermally developing or fully developed, with the constraints of porous medium – its material properties, location, shape, size, etc., flow and boundary conditions under which the simulation is conducted.
(IV) Variable Porosity (VP)
In the aspect of porosity, numerous works consider various problems of flow and heat transfer through a constant porosity medium [Beckermann & Viskanta, 1987; Nield et al.,
1996; Poulikakos & Kazmierczak; 1987; Kim et al., 1994; Kaviany, 1987; Nakayama et al., 1990; Ould-amer et al., 1998]. But porous systems with variable porosity near the
78 bounding walls have been shown in a number of experimental studies [Okuyama et al.,
2000; Sederman et al., 1997] that variable porosity plays a vital effect in velocity field and results in flow accelerating at the region next to the impermeable bounding walls.
The resulting effects of variable porosity on the pressure drop and heat transfer have been studied by Vafai et al. [1985], Reken& Poulikakos [1988] and Hunt & Tien [1988a,
1988b]. In this thesis, Chapter 2 has introduced the methodology that solid phases in porous matrix can vary in size, shape and location. It also can be formed in a random manner or in a structured pattern. Variable porosity is inherent in the current study.
(V) Thermal Dispersion (TDP)
The term “thermal dispersion” is used to refer to the thermal transport enhancement occurring when fluid undergoes mixing as it traverses the tortuous paths around the solid phase in a porous medium. It is the result of hydrodynamic effect and its relative magnitude is high compared to molecular diffusion at higher Reynolds numbers, especially when the heat conductivity of solid phase is low. In the past, several studies have been attempted to quantify dispersion in pack beds and open cellular metal foams, experimentally and numerically [Calmidi, 1998; Hunt & Tien, 1988 a & b; Hsu & Cheng,
1990; Lage & Narasimhan, 2000]. Most of them provided the validation data from experiments. One of the difficulties in performing a direct measurement in a porous medium is the limited access to the void spaces within it. Instead, experiments usually provide only indirect results by measuring the thermal energy in and out of the medium.
Then the thermal dispersion model is calibrated/postulated by subtracting off the energy
79 transfer caused by molecular conduction and advected enthalpy changes obtained by solving volume averaged energy equation from the measured net energy flux between porous medium [Niu and Simon, 2004]. Consequently, proposed dispersion models vary widely and do not yield consistent results.
A summary of the numerical simulation studies presented above is offered in Table 5.
According to the experimental researches conducted in recent years by using non- intrusive measurement techniques, their findings all point to the importance of considering “heterogeneity” in the porous media and the resulting mass, momentum, and thermal transports in the flowfield. Part of this chapter is devoted to numerically exploring the effects caused by heterogeneity which is often neglected in the past numerical simulations. The interplay of viscosity, conductivity and convection forms a complex system. In the section of results and discussion, results are presented and discussed in terms of the conventional parameters, including the local Nusselt number,
Reynolds number, Prandtl number, Péclet number, local effective heat conductivity and others deduced from their combinations. The importance and effects of these parameters will be discussed along with the case studies.
Assumption References Formulation and Model
Homogeneous fibrous Angirasa & Peterson, Volume averaged material, local thermal 1999; equations of mass, equilibrium between solid momentum, and energy and fluid phases, high solved for 2D channel Reynolds number, flow; various finite parallel-plates channel difference methods. Fully developed channel Poulikakos & A modified Brinkman
80 flow, local thermal Renken, 1985 momentum equation equilibrium, constant solved after thermo-physical linearization, then properties, variable single state energy porosity equation solved by Keller Box method VP. Fully developed channel Kaviany, 1985 Volume averaged flow, local thermal equations of mass, equilibrium, variable modified Darcy porosity, axial conduction momentum equation negligible, constant and a single state properties energy equation; finite difference method used, 2D channel flow. (VP) Homogeneous material, Nield et al., 2002 Brinkman momentum no axial conduction (high equation, and two- Péclet number), thermally temperature volume developing, local thermal averaged energy non- equilibrium equations for 2D channel flow with isothermal walls; Graetz method used. (LTNE+TD) Variable porosity, local Alzmi & Vafai, 2002 Modified Brinkman thermal non-equilibrium momentum equation and between solid and fluid two-temperature volume phases, thermal dispersion averaged energy equations considered; properties for 2D channel flow isotropic, homogeneous (TDP+VP+LTNE) and constant Variable porosity, thermal Hsu & Cheng, 1990 Ergun momentum dispersion considered, model, single phase parallel-plates channel volume averaged energy equation including semi- empirical odes for thermal dispersion (VP+TD) Fully developed channel Hunt & Tien, 1987; Volume averaged flow, constant constitutive Calmidi & Mahajan, equations of mass, property, thermal 2000 modified momentum dispersion considered, equation, and single high porosity fibrous energy equation (TDP) foam. Homogeneous material, Narasimhan et al., 2001; Forchheimer-extended constant thermal properties Nield et al., 1999. Darcy momentum
81 with temperature dependent equation,(VT) viscosity, local thermal equilibrium, no axial conduction, fully developed flow, Current study 3D Navier-Stokes equations for fluid and heat conduction equation for solid (VP+LNTE+VT+TDP+TD)
Table 5: Summary of numerical simulations of forced convective in porous media.
Reading from summarized review of past works in Table 5, it is evident that progress has been made in establishing a knowledge base of convective heat transfer pertinent to the use of porous media in broad applications. However the knowledge base is by no means complete and research needs are still extensive. It is also evident that there is a strong need for developing numerical capabilities to accurately predict heat transfer effects for various real-world geometric designs and materials of solid matrix used in porous media.
Thus, the study conducted in this thesis contributes, to the best of the author’s knowledge, for the first time in providing a detailed computational analysis of fluid and heat transport processes for a broad range of porous media. The simulations are intended not only to study thermal dispersion, local non-equilibrium effects but also to include heterogeneous state-dependent thermo-physical properties of fluid phase.
In addition to these aspects described in the beginning of this section, the effective heat conductivity is widely used as an averaged bulk property for numerical models [Hsu &
Cheng, 1994; Bauer, 1993] in the studies of porous media. Empirical formulations are
82 formed by correlating with experimental data to account for the spatial heat conduction in packed sphere bed [Sahraoui & Kaviany, 1993; Carbonell & Whitaker, 1984]. Section
4.3.1.2 is devoted to study the local thermal conductivity of fluid phase in dimensionless parameters of the ratio of local averaged thermal conductivity to the inflow thermal conductivity and Reynolds number. Results are tabulated with respect to different solid phases considered in Table 1 of Chapter 1, together with theoretical results.
Before the results of numerical simulations of porous media are presented and discussed, related conjugate heat transfer cases are presented. A widely used single plate-plate forced convection model is studied under various conditions. The resulting Nusselt number, a dimensionless heat transfer parameter, is compared with theoretical results.
Following the validations of solid-fluid interaction in forced convection, the study of forced convection in porous media is described in which the numerical setup is similar to the one described in Chapter 3. A parametric study of dispersion effect, thermo-physical properties (heat conductivity and heat capacity) of solid phases listed in Table 1, and mass flow rate of fluid phase is conducted. The resulting velocity and temperature fields are reported and discussed. Quantification of these effects will be presented in terms of
Nusselt and Péclet numbers. Definitions of these dimensionless parameters used in the following sections are expressed as,
∗ Tx()−T()x T = w (4.1) Txmw()−T()x and
qxw ()H Nux = (4.2) Kwm(x)[Tx()−Tw()x]
83 where T* is the invariant temperature, Tm is the volume-averaged temperature, and Nux is the local Nusselt number. The Péclet number (Pe) is the ratio of convection effect to conduction effect and is the product of Reynolds number and Prandtl number whereas the
Prandtl number (Pr) is the ratio of momentum diffusivity to the thermal diffusivity.
4.2 Heat Transfer Validation Cases – Forced Convection in Single
Plate-Plate Channel
A test case of conjugate heat transfer was used to address the heat transfer validation study. A simplified model of an electronic board cooled by a laminar flow in a parallel- wall channel of fixed length of L is used. The channel walls are either isothermal or adiabatic. The board substrate with a thickness of t is on the top, and the coolant fluid flows below in the channel. The coolant used in this study is air. Figure 4.1 is the schematic description of this forced convection problem. The applications of this model and its kind have been studied extensively by numerical, theoretical and experimental approaches [Bejan, 1995; Vargas et al., 2001: Morega et al., 1995; Incropera, 1988;
Peterson & Ortega, 1990].
84 Circuit board, Solid Substrate Tw
t
Tin coolant Insulated/ Tw Uin
L
Figure 4.1 Schematic cooling of electronic board with a substrate between coolant and board.
Cases studied include the use of two types of substrates and two kinds of boundary conditions of the bottom plate. If a stack of parallel plates (e.g. fins of heat sink, printed circuit boards) are used as cooling device, the bottom plate is considered as an inner plate representing a simplified model of the stack. Then, the boundary condition of the lower plate is isothermal. Simulations also cover the situation of one-sided heating in which the bottom plate is an insulated external plate and adiabatic condition is imposed. Also, two types of substrate are used in this study of forced-air cooling in single plate-plate channel.
The first is the use of silicon rubber which is broadly used as an insulation material. The other substrate is made out of aluminum which is highly conductive. Air is used as a coolant. And, the physical dimensions shown in Figure 4.1 are: L =10 cm and t = 0.5 cm.
The uniform inflow and boundary conditions are: uin = 1.59 m/s, Tw = 900 K, and Tin =
300 K. The resulting Re based on hydraulic diameter is 2000 and the flow is assumed
85 laminar [Bejan, 1995]. The temperature, velocity contours, and the local Nusselt number are presented and discussed.
4.2.1 Bottom Plate as an Inner Plate
Figures 4.2(a) and (b) show the results for silicon rubber substrate in terms of velocity and temperature contours respectively. The momentum boundary layers depicted from velocity contours in Figure 4.2(a), and thermal boundary layers from temperature contours in Figure 4.2(b) indicate that flow is fully developed at the exit of the channel.
The results for Pr ≈1 of air also reflect in the sizes of hydrodynamic entrance length and thermal entrance length being of the same order.
In Figure 4.2(b), the heat generated from the circuit board was conducted away from the top plate and through the silicon rubber substrate. Then at the surface between coolant and substrate, the heat was convected away from the substrate. The lower plate is kept at constant temperature, since it is an inner plate of a stack of parallel plates (e.g. fins of heat sink, printed circuit boards).
86 u: 0.002 0.012 0.022 0.032
(a)
T: 350 450 550 650 750 850
(b)
Figure 4.2: Silicon rubber substrate with isothermal walls at 900 K, (a) velocity contours, (b) temperature contours.
The resulting heat-transfer distributions on both top and bottom walls, expressed in terms of local Nusselt number, Nux, are plotted in Figure 4.3; the averaged value between that of two walls is also included, whose value at the exit is about 7.74. The heat transfer result will be compared with the one from Aluminum substrate later.
87 45
Substrate - Silicon Rubber 40 top wall 35 lower wall averaged 30
25 Nu 20
15
10 7.74 5
0.2 0.4 0.6 0.8 1 1.2 x/L
Figure 4.3: Local Nusselt number Nux versus dimensionless x of heat transfer at the contact surface of coolant and substrate of silicon rubber.
Different from the low heat conductivity substrate, the heat generated by the circuit board is conducted instantly across the aluminum substrate. In Figure 4.4(b), the temperature contours indicate that the substrate reaches a constant temperature. This is due to its high thermal conductivity. Figure 4.4(a) demonstrates the flow accelerates more rapidly with the aluminum than with the silicon substrate and Figure 4.4(b) shows a thicker thermal boundary layer. In addition, the pressure drop across the channel is about 50% more than with the rubber substrate. See Appendix C for detailed values.
88 u: 0.002 0.012 0.022 0.032
(a)
T: 350 450 550 650 750 850
(b)
Figure 4.4: Aluminum substrate with constant wall temperature, Tw=900K, (a) velocity contours, (b) temperature contours.
Similarly, the heat transfer rate at wall is also presented in the plot of Nusselt number along the axial direction in Figure 4.15. It suggests that aluminum provides a better heat transfer performance but at the cost of higher pressure drop.
89 25 Substrate - Aluminun 20 top wall lower wall 15
10 Nu
5
3.665
0 0.2 0.4 0.6 0.8 1 x/L
Figure 4.5: Local Nusselt number Nux versus dimensionless x/L, at the contact surface of coolant and substrate of aluminum.
4.2.2 Bottom Plate as an External Plate
Another case considers that the lower plate is an external plate and insulated. Therefore, the adiabatic boundary condition is imposed on the lower wall. Similar to the preceding case, plots of velocity and temperature contours are presented in Figure 4.6 and 4.7 for substrate of silicon rubber and aluminum respectively. A thinner thermal boundary layer is seen in the aluminum case than the silicon rubber one, by comparing Figures 4.6(b) and 4.7(b). For both substrates, the resulting dimensionless heat transfer rates at the top
90 are presented wall in terms of Nux in Figure 4.8. Obviously, the metal outperforms the rubber in cooling.
The resulting heat transfer coefficients in Nux are plotted against the dimensionless x starting from the entrance. In both cases of aluminum substrate, the board temperatures are almost constant and uniform due to the high heat conductivity as indicated in Figure
4.4(b), and 4.7(b). The corresponding Nu at the exit are 3.66 and 2.99 respectively, see
Figures 4.4 and 4.8. Both have achieved good agreements with the values derived from theoretical studies of forced convection in two infinite parallel plates. They are 3.77, for the case of keeping both plates at constant temperature and 2.43, for the case of the top wall at constant temperature while the lower wall insulated [Shah & London1978].
91 u: 0.002 0.012 0.022 0.032
(a)
T: 350 450 550 650 750 850
(b)
Figure 4.6: Silicon rubber substrate with Tw = 900 K and the other parallel plate insulated (a) velocity contours, (b) temperature contours.
92 u: 0.002 0.012 0.022 0.032
(a)
T: 350 450 550 650 750 850
(b)
Figure 4.7: Aluminum substrate with isothermal top wall at Tw = 900 K, and the lower parallel plate insulated: (a) velocity contours, (b) temperature contours.
93 35 30 Substrate 25 Aluminum 20 Silicon Rubber 15
10 Nu
5
2.99
0 0.2 0.4 0.6 0.8 1 x/L
Figure 4.8: Nux along longitudinal direction x at the contact surface of coolant and substrates of silicon rubber and aluminum.
4.3 Numerical Setup for Forced Convective Heat Transfer in Porous
Media
The numerical procedures are set in the manner as described in Section 3.3. Heat is provided from outside. Two sets of heat sources will be considered, respectively denoted as
Case (a): Th = 600K and Tc = 300K, (4.3)
94 Case (b): Th = Tc = 600K. (4.4)
Case (a) is heated on one-sided, resulting in an unequal heating, which can occur in the situation of the wall being a heating element; Case (b) has heated surfaces on both walls.
4.3.1 Two-Dimensional Channel
A 2-D channel of physical dimensions 20 cm x 10cm is considered, same as the channel used in Chapter 3. The porous sample has a length of 20 mm and is situated in the center of the channel. The schematic is shown in Figure 4.9. An example of the structure of pore distribution and pore scale, as used in the study, can be seen in black color in Figure
4.11(a), which will be further discussed later.
Tc
Fully developed flow Flow exit
Porous Medium
y x ∆1 ∆2 ∆3 ∆4 ∆5
Heated Wall , Th
Figure 4.9 Schematic of two-dimensional porous channel with heat source from below.
95 The computational domain is chosen to be long enough that flow is fully developed before entering the porous region. A structured mesh is adopted here for its better performance (accuracy and convergence) in dealing with boundary layer. There are
60,000 elements used in computation. The same channel case which has no porous sample confined between two parallel plates is also computed with the same boundary conditions for comparison. It is referred to as the baseline case in the following. Again, the computed results are sampled and presented at five locations in the X-direction, as
indicated in Figure 4.9. They are denoted by ∆1, ∆2, ∆3, ∆4, and ∆5 and defined at
locations: x=0.05 m, 0.09 m, 0.1 m, 0.11 m, and 0.15 m respectively. ∆1 is located just
upstream of the porous block, while ∆5 just downstream of the block. The entrance,
middle plane and end of the porous medium are labeled as ∆2, ∆3, and ∆4 respectively.
Several different solid materials are analyzed. First the solid material has the thermal property of aluminum with a heat conductivity of 218 W/m-K and other values as listed in Table 1 of Chapter 1. Since it is not possible to accurately quantify dispersion effects when solid-phase conductivity is much larger than its fluid phase counterpart, a hypothetical material having the same conductivity of air, 0.0262 W/m-K, is used. The aim is to study how the structure of porous medium plays a role in thermal transport. In other words, the dispersive effect becomes the dominant mode of transport if the conduction through the solid phase and interfacial convection are omitted. Numerical studies are performed in the range of non-dimensional parameters, Red ~ 0.5 -100, Pr ~
0.7 – 1.5. A typical inflow velocity is in the range of 0.05 m/s to 1.0 m/s. At the inlet, the fluid enters with a specified mass flow rate and temperature, m and Tin respectively.
96 4.3.1.1 Result and Discussion
Results are presented and discussed in terms of conventional macroscopic parameters or effective bulk properties. A baseline case is computed to provide results of a plain channel flow, as described above. It is also used to compare against the results available in the literature [Shah & London, 1974; Heaton, et al., 1964]. In all cases, the agreement is good. Two combinations of heating situations are considered with various isothermal wall conditions, Equations (4.3) and (4.4). Four porous materials which have been described in Section 1.4 are used as an insert into the channel. Flow details are given for velocity, temperature, and heat transfer rate at the heated wall of a representative material, Silicon Nitride (θ = 954). Results from the other three materials are obtained and compared against that of the baseline, Silicon Nitride porous systems in the following parametric study. Further analyses are conducted to allow comparisons with several theoretical and semi-empirical models in estimating effective heat conductivity, see
Section 4.3.1.2.
The hypothetic solid material of air is used to study the thermal dispersion. It is noted that no heat conduction occurs between the fluid and solid phases because they both have the same thermal properties and the same temperature at the contact surface. Hence, the temperature field is expected to reveal information at pore scale in thermal dispersion.
The analysis will be given in Section 4.3.1.3.
97 (I) Baseline Case
In this study, the velocity, temperature profiles and Nusselt number are determined for flow through the channel bounded by isothermal parallel plates. The dimensionless
temperature profile, T* versus y*, at locations defined at ∆1, ∆2, ∆3, ∆4, and ∆5, are plotted in Figures 4.10(a), and (b). Figure 4.10(a) shows the temperature contours and Figure
4.10(b) is the temperature profile described at different physical locations. When heat transfer begins at the inlet of the channel, the velocity and temperature profiles develop simultaneously. Figure 4.10(b) is plotted in the conventional form which is an invariant against x for finite channel length. Apparently the thermal boundary layer at x/L of 0.75 is still developing and approaching a fully developed profile, as indicated in Figure
4.10(b). It has been known that for any finite length and fully developed field, an invariant temperature and an invariant local Nusselt number exist [Kays & Crawford,
1980]. Figure 4.10(c) shows the non-dimensional velocity profiles in the direction of y, whereas x/L represents the non-dimensional x. The profiles are asymmetrical because only the lower wall is heated. Fluid close to the heated wall forms a thicker momentum boundary layer because the air viscosity increases with temperature. Figure 4.10(d) plots the distribution of Nux along the wall starting at the inlet of the channel (x/L=0) for Case
(a). Note that this baseline case is used as a reference to illustrate how flow changes in porous medium.
98
Figure 4.10 (a): Temperature contours of the nonporous channel with the background showing the mesh used.
Figure 4.10(b): The invariant temperature profiles along the direction of y in the channel at various sampling positions along the axial coordinate x.
99 Figure 4.10(c): Velocity profiles along the transverse direction at all sampling locations. The asymmetric velocity distributions in a symmetric channel are due to the one-sided heating.
20
15 Nu
10
5 3.82
0 0.2 0.4 0.6 0.8 1 x/L
Figure 4.10(d): Local Nusselt number Nux versus the longitudinal coordinate x of the baseline case. All results are obtained from the same channel as depicted in Figure 4.9 but without the porous insert.
100 (II) Porous System of Silicon Nitride and Air
Next, the case of channel flow with the porous region positioned in the middle, as shown in Figure 4.9, is considered. The same boundary and inflow conditions as the previous case are applied here also. In this case, the porous medium consists of Silicon Nitride solid matrices.
Figure 4.11(a): Temperature contours of the porous sample in a channel, along with the mesh used and the porous region in which the solid matrices are shown in black. Note that results of subfigures (a)-(d) are obtained from a porous system of air and silicon nitride (θ = 954) with porosity ε = 0.83.
101 Figure 4.11 (b): The invariant temperature profiles along the direction of y* in the channel are plotted for various sampling positions. The disrupted growth of thermal boundary layer can be noticed from the collapsing of lines at ∆1, ∆2, ∆3, and ∆4.
The qualitative and quantitative descriptions of the thermal boundary layer inside the porous region are displayed in Figures 4.11(a) and (b) respectively. Figure 4.11(a) clearly depicts that the thermal boundary layer begins to thin down as the fluid is entering the porous block and grows back after leaving it. This behavior of boundary-layer thinning has also been observed in experimental investigations and discussed in Renken &
Poulikakos [1987] and Amiri & Vafai [1994]. The distributions in Figure 4.11(b) are plotted in the conventional form, representing the dimensionless temperature distributions across the channel at various x-stations. They are taken within the porous insert at the entrance (x/L = 0.45), middle (x/L = 0.5), and exit (x/L = 0.55) planes. It is remarkable that in Figure 4.11(b) the thinning of thermal boundary layer results in the collapsing of the dimensionless temperature profiles inside the porous region into a nearly invariant profile. This enhancement of heat transfer and a larger temperature gradient inside the
102 porous region may cause some concern in practice for the integrity of matrix structures due to excessive heating.
The velocity profiles are displayed in Figure 4.11(c); the locations with zero or small velocities correspond to the interfaces of solid matrices. The spikes are due to the heterogeneity of the porous block, thereby affecting deceleration/acceleration of the fluid as it flows through the torturous path. By contrary, no collapsing of velocity profiles should be expected because they are essentially local phenomena, depending on the composition of solid matrices. These profiles are seen to exhibit differences at the pore scale. Nevertheless, it is interesting to notice that the velocity profiles after leaving the porous region (x/L=0.75 and 1.0) return to a smooth profile nearly immediately, but the history from the porous region is also evident because their shapes resemble the averaged velocity in the porous region (x/L=0.45, 0.5, and 0.55). This behavior is consistent with the observation by Kaviany [1985]. This non-uniformity in velocity distributions is expected to affect the heat transfer rate from or to the wall. As an example, the axial heat transfer rate at the heated wall is plotted in Figure 4.11(d). The drastic increase in Nux in the porous region is evident, but the relative magnitude of this enhancement in heat transfer is clearly determined by the morphology of the solid matrix. The Nusselt number at the exit of the channel exhibits a 67% increase from the baseline case, solely due to the existence of a 2mm thick (only 2% of the channel length) porous medium of silicon nitride.
103
Figure 4.11 (c): Profiles of non-dimensional x-component velocity along the transverse direction at various sampling locations. V' is nondimensionalized by uin and y' is same as y*.
25
20
15 Nu
10
5 4.90 0 0.2 0.4 0.6 0.8 1 X/L
Figure 4.11(d): Plot of local Nusselt number Nux versus the longitudinal coordinate x.
(III) All Porous Systems Listed in Table 1
104
Following the previous detailed pore-scale displays of hydrodynamic and heat transfer characteristics, this subsection will include all porous systems studied in this research. To provide a systematic investigation of transport phenomena in porous media, a parametric study with respect to various porous media, has been performed. The resulting fluid flow and heat transfer relationships are presented in the rest of this section and discussed in terms of the conventional parameters, including the local Nusselt number, Prandtl number, Péclet number, Reynolds number and the local effective heat conductivity.
0.4
Kt_s = 218 W/mK (a) (b) 0.3 (c) (d) (e) kt _s = 0.02616 W/mK (a) (b) H 0.2 (c) y/ (d ) (e)
0.1
0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 f
Figure 4.12: The ratio, f, of local heat conductivity of fluid phase to that at the inflow condition (300K and 1atm) in terms of the distance away from wall at various sampling locations, where (a) denotes location at ∆1, (b) at ∆2 , (c) at ∆3, (d) at ∆4, and (e) at ∆5 respectively. Note that f becomes unity in the core region.
105 Figure 4.12 shows the distributions of a non-dimensional local heat transfer factor, f, which is defined as a ratio of local heat conductivity to that at the inflow condition, along the distance from the heated wall of the channel. This figure shows the thicknesses of thermal boundary layers with two solid materials. The closed symbols, clustered near the wall, show that the porous system of aluminum/air has a much thinner thermal boundary layer than that of the hypothetical porous system. In Figure 4.12, the red and purple lines
denote the results at ∆1 , significantly upstream of the porous region, and they are seen to be nearly coincident with each other. This is to be expected because the flow is still far from the porous insert and should not be affected by it. Other lines representing locations at (b), (c), (d) and (e) significantly depart from each other between the two solid matrix materials, having heat conductivities of 218 W/mK and 0.0262 W/mK respectively.
Figure 4.13 summarizes the heat transfer rate at the isothermal (Case (b), Equation (4.4)) bottom wall of the channel for four different types of porous media. Also included is the baseline case as a basis to quantify the degree of heat transfer enhancement due to the flow blockages in the porous region. Taking together both Figure 4.12 and Figure 4.13, one can reach the following conclusions. It is evident that the thickness of the thermal boundary layer decreases and the heat transfer rate increases as the heat conductivity
K of the solid matrix increases. The latter, however, is related to the former because the thinning of thermal boundary layer also results in a higher heat transfer rate. It is seen in
Figure 4.13 that in the case of aluminum matrix, the heat transfer rate can be enhanced by as high as forty times inside the porous block. Moreover, the maximum Nu attained in each solid material increases with the heat conductivity. This thinning/disruption in the
106 growth of thermal boundary layer are also observed for other types of solid phases, albeit with different magnitudes.
It is of interest to note from Figure 4.13 that heat transfer rate begins to depart from the baseline at a distance on the order of 0.3L upstream of the porous block, which is far greater than the characteristic length of the boundary layer. Thus, this upstream effect is clearly not felt through the boundary layer. It is likely through a longitudinal acoustic wave which is associated with a length scale of L. In addition, the heat transfer rate downstream of the porous block, as seen in Figure 13, has an interesting behavior in which (1) the solids with moderate Kt (say ≤ 25) have reached a minimum value of Nu at the exit of the block and then relaxes to an asymptote far downstream, (2) the curves of these low to moderate Kt practically fall on a single curve after leaving the porous region, and (3) the curve of high Kt has never reached a minimum before relaxing to an asymptote. As indicated previously, the spikes inside the block are attributable to the heterogeneity of the solid structure and the distribution should vary from a manufactured porous product to another even they are made of same material.
107 102 Baseline Kt_solid = 0.0261 Kt_solid = 0.166 Kt_solid = 25.0 Kt_solid = 218.0 Nu 101
100 0.2 0.4 0.6 0.8 x/L
Figure 4.13: Distributions of local dimensionless heat transfer along the bottom wall of four solid phases, the upward spikes indicate the enhancement of heat transfer contributed from the physical blockages of porous media. All plots are obtained for both parallel plates kept at a constant temperature of 600 K, Case (b), at Re = 1,100.
Corresponding to the same conditions in Figure 4.13, Figure 4.14 plots distributions of
vorticity at location ∆3 which is right in the middle of the porous region. Note that the vorticity is non-dimensionalized by uin/H. It is clear that the mechanism of generating vorticity, which is indication of spatial variations (gradients) of velocity, is primarily due to the geometric characteristic of porous medium considered, not directly by the physical properties of fluid. In the baseline case, a straight open channel without porous region,
108 the heat and momentum are transferred by molecular diffusion and there is little vertical motion in the boundary layer to generate vorticity, as shown in red color in Figure 4.14.
1
Baseline 0.8 Hypothetical solid (air) Silicon Rubber Aluminum 0.6 y* 0.4
0.2
0 0 20 40 ω∗60 80 100 120
Figure 4.14: Vorticity distribution along the dimensionless transverse direction, y, on the mid-plane of the porous medium (∆3). The spikes show the enhancement of vorticity caused by the solid matrices inside the porous region. The locations of spikes correspond to the locations of solid matrices.
Its magnitude vanishes away from the wall, in the “core” region. On the other hand, the vorticity generated by solid matrices is enormously larger and random in nature. The irregular nature is solely determined by the structure of the porous medium. Interestingly,
109 Figure 4.14 reveals that the least conductive solid material in this study, the hypothetical solid/air system, generally exhibits a larger or equal vorticity than the others of higher conductivity. From Figure 4.14, it is concluded that the vorticity generated in the porous region has little connection with the thermophysical property of solid matrices. Figure
4.15 also displays the distribution of vorticity with respect to the change of mass flow rate (or Reynolds number) in the case of hypothetical solid/air system. It clearly shows that Reynolds number has no or little effect on the vorticity distribution. For the same porous system shown in Figure 4.15, Figure 4.16 plots its heat transfer rate in terms of
Nusselt number with respect to various Reynolds number. Note that the heat transfer rate increases with the Reynolds number. Reading from Figure 4.15 and 4.16 together, it indicates the heat transfer enhancement may not be caused by the vortices occurring in the porous medium. However, since vorticity has a direct influence on fluid mixing, the study of porous flow will be of practical importance in this regard. In what follows, a further study on the effects of vorticity in porous medium is conducted and its relationship with thermal dispersion is discussed. Further analysis on thermal dispersion will be provided in Section 4.3.1.3, with the help of using the same hypothetical material.
At least, it is evident that the vorticity generated throughout the porous region is mostly a function of local porous matrix structure – location, shape, and size, but insensitive to heat conductivity of solid phase (K in Figure 4.14) as well as mass flow rate of fluid phase (Re in Figure 4.15).
110 1
Re 550 0.8 820 1,100 2,200 5,500 0.6 y* 0.4
0.2
0 0 50 ω∗ 100 150
Figure 4.15: Vorticity distributions on the center plane of the hypothetical solid/air (system of θ = 1.0) under Case (b). Re is based on the inflow velocity and channel height.
111 50 40 Re
30 5,500 2,200 1,100 20 800 550
10 Nu
0.2 0.4 0.6 0.8 1 x/L
Figure 4.16: Local dimensionless heat transfer rate along the bottom plate at various Reynolds numbers. The porous system studied is composed of air and the hypothetical solid/air as the solid phase, i.e. θ = 1.
It is known that Prandtl number Pr is insensitive to temperature in gases made up of
simple molecules because their structure is least responsive to temperature changes.
The air is made up of diatomic gases (N2 and O2) and Pr = 5/7 is accurate over a
moderate range of temperatures. Plots in Figures 4.17 (a)–(c) are the local Prandtl
number profiles of fluid phase within the thermal boundary layer for the hypothetical
material; they all lie in a limited range between 0.7 and 0.72 at three planes inside the
porous region for three Reynolds numbers, shown respectively in subfigures (a)-(c).
112 They further confirm the insensitivity of Pr to changes in temperature. This also
suggests that viscosity and heat conductivity of the fluid phase (air) vary similarly in
the temperature range considered, as consistent with the result from the kinetic theory
of gases.
y*
0.4 0.4 (a) (b) 0.3 0.3
0.2 0.2
0.1 0.1
0 0 0.7 0.705 0.71 0.715 0.705 0.71 0.715 Pr Pr 0.4 (c) 0.3
0.2
0.1
0 0.705 0.71 0.715 Pr
Figure 4.17 (a) - (c): The local Prandtl number profiles along the distance away from the heated wall at various x* locations, where — 0.25 (∆1), — 0.45 (∆2), and — 0.5 (∆3) —
0.55 (∆4) , and — 0.75 (∆5), of (a) Re = 1,100 (b) Re = 2,200 and (c) Re = 5,500.
113 0.4 (a) 0.4 (b) Re
1,100 0.3 5,500 11,000
0.2 0.2 y* y*
0.1
0 0 1.2 1.6 1.2 1.6 f f (d) (c) 0.4
0.3 0.3
0.2 0.2 y* y*
0.1 0.1
0 0 1.2 1.4 1.6 1.2 1.4 1.6 f f
(e) 0.4
0.3
0.2 y*
0.1
0 1.2 1.4 1.6 f Figure 4.18 (a) - (e): Profiles of non-dimensional effective heat conductivity at five planes, ∆1-∆5, respectively shown in (a)-(e), for three Re=1,100 (─), 5,500 (─), and 11,000 (─).
114 Figure 4.18 shows the profiles of non-dimensional local effective thermal
conductivities (f = [Kf]eff/ [Kf]in) of fluid phase in the thermal boundary layer. Note that
Keff is largest at the wall and returns to the value in the core region, i.e., heat
conductivity of air at the inflow condition. Also the profiles do not seem changed by
the presence of solid matrices.
26
24 baseline (Re = 1,100) K : air, Cp : 2.4xair (Re = 1,100) 22 K : air, Cp : air (Re = 1,100) baseline (Re = 2,200) 20 K:air,Cp:2.4xair(Re=2,200) 18 K : air, Cp : air (Re = 2,200)
16
Nu 14
12
10
8
6
4 0 0.2 0.4 0.6 0.8 1 x/L
Figure 4.19: Local Nusselt number along the bottom wall, showing the effects of Re, heat conductivity and specific heat of solid phases.
115 For the case of the hypothetical solid/air material (θ = 1), Figure 4.19 shows that higher velocities (higher Re) produce higher Nu, implying more thermal energy being carried away from the wall compared with that by conduction. This is due to the fact that a higher Re results in a thinner thermal boundary layer, thereby leading to an increase in
Nusselt number. Note that all Nu resulting from the porous region under two Re and two different values of specific heat CP are consistently higher than that of the baseline case where no porous region is inserted. However, the effect of Cp, change by a factor of 2.4, does not seem make a change of similar order of magnitude. But, the effect of Cp in porous medium may be more pronounced in transient flow [Okuyama & Abe, 2000].
In conclusion of this subsection, all analyses performed with thermophysical property of solid phase, K and Cp, and flow rate of the fluid phase, in terms of Re, indicate that the enhancement of heat transfer in the porous system is mainly attributed to the thinning of the thermal boundary layer within porous media. This behavior changes the gradient and fluctuations of the temperature field. Meanwhile, the interplay of velocity and temperature fields caused by the solid matrices gives rise to distorted velocity and temperature profiles downstream of porous media. It is interesting that for all Re, Nu drops rapidly to a minimum near the exit of the porous region and slowly relaxes to the respective equilibrium values, which are dependent on Reynolds number.
116 4.3.1.2 Effective Thermal Conductivity
In the study of heat transfer in the porous medium, the effective heat conductivity is considered the relevant parameter and has been intensively studied since the pioneering work of Maxwell [1891]. Kaviany [1991] has provided an extensive review on the subject and discussed the ranges of applicability of available models in the literature. The geometric average model is often used for estimating effective heat transfer [Angirasa &
Peterson, 1999] based on the thermal conductivity of the constituent phases. A simple similarity is drawn between effective thermal conductivity of a composite and resistors of an electrical circuit. Hence, a first-order model of the effective heat conductivity of a fluid-filled porous medium in the following expression
K(eff =−1ε )Ks +εKf (4.5)
is commonly adopted. This model is made simply by accounting for the volume fraction of each substance and its corresponding heat conductivity. Some other more sophisticated models are derived through rigorous mathematics. To make these theoretical models more useful, extensions have been proposed by Verma et al. [1991] and Misra et al.
[1994] through adding correction terms adjusted by fitting with available experimental data. Other empirical models have been postulated from experiments for packed beds or granular media include Nozad [1985], and Hsu & Cheng [1994]. Unfortunately, none are found to be universally satisfactory.
117 Recently new models have been introduced for the cases of highly conductive metallic solids. For example, Calmidi & Mahajan [1999] and Boomsma & Poulikakos [2001] developed models utilizing geometrical estimate for calculating the effective thermal conductivity of porous systems with constituent metallic and fluid phases. Calmidi &
Mahajan proposed a one dimensional heat conduction model of a two dimensional array of hexagonal cells representing the porous medium. Later Bhattacharya et al. [2002] extended the Calmidi & Mahajan model with a circular intersection, which results in a six fold rotational symmetry. On the other hand, Boomsma & Poulikakos used tetrakaidecahedron cells with cubic nodes at the intersection of two nodes to model highly porous metal foams. These geometrical models are based on experimental data to provide the geometric parameters.
Instead of making simplifying assumptions about the solid phase, whether its shape, size, or location, this thesis allows a random geometry at the pore scale, hence is more relevant to reality. For validation purposes, an effort is made in this section to connect the current results with the conventional macroscopic averaged property, effective heat conductivity.
This averaged bulk property is limited to its applicable ranges because they are rooted on the specific experimental results. While the current approach possess a complete description of the flowfield at the pore scale, e.g., temperature T(x,y) in two dimensions, certain averaging of the computed data is performed in order to facilitate the comparison with theoretical model or experimental data, both of which are carried out at the macroscopic level. In the following, the effective heat conductivity of fluid phase (Kf)eff
(x) is defined as the averaged value of the local heat conductivity of fluid and is obtained
118 by integrating Kf (x,y) along the transverse direction from wall to the edge of thermal boundary layer and is divided by the area covered. Furthermore, a systematic study of transport behavior with respect to various porous systems has been performed. The ratio of [Kf]eff (x) to the fluid conductivity at inflow condition ([Kf]in), i.e. the value outside of the thermal boundary layer, for two representative porous systems are plotted in Figure
4.12 in previous section. The model of Calmidi & Mahajan [1999] is chosen for comparison; its expression is written as
bb3 b 33rr(1−−)3(3) 2 LL2 L(−1) (K ) = []++ (4.6) f eff bb b 3 (2−−)K (3 2 )K (3 3 −4r )K LLffLf
Note that for the effective heat conductivity of solid phase, Kf is substituted with Ks.
Equation (4.6) is based on a two dimensional array of hexagonal cells but is correlated with experimental results. Therefore, it is postulated for three dimensional metal foam having large pores and fine connecting ligaments. The geometric parameters in the above equation are (b/L, r), where b and L represent the half thickness and half length of bump, r the half thickness of fiber. Following the definition of Calmidi & Mahajan, the geometric parameters used in this study (Figure 4.9) correspond to values of (b/L, r) =
(0.47, ≈0.0). Also, their study is restricted to pure conduction in a stagnant fluid flow condition and excludes convection. The ratio of the effective heat conductivity to the
fluid conductivity, defined as f = [Kf]eff /[Kf]in, is calculated and given in Tables 6 and 7 for various porous systems. To enable a comparison between the present results with that of Calmidi et al.’s geometric model, Table 6 lists [Kf]eff of the hypothetical solid/air material at very low speed to minimize any effect from convection. In Table 6, at the exit
119 of porous medium, ∆4, the computed result agrees well with the prediction from Equation
(4.6), whose values are given in the parenthesis. Since the theoretical estimates, given in the parenthesis, are based on periodic geometric patterns, they do not recognize the spatial variation of f, thereby resulting in same values at different locations, ∆3 and ∆4.
See also Tables 7 and 8.
Solid Κ (W/K-m) f @∆1 f @∆3 f @∆4 f @∆5
Air 0.0262 1.1559 1.1594 (1.147) 1.141 (1.147) 1.162
Table 6: The effective heat conductivity of fluid phase at Re = 270. The values included in the parenthesis are from Equation (4.6).
In the study of effective heat conductivity in forced convection in porous system, the question as to whether convection is negligible in deriving effective conductivity arises.
It is useful to observe the effects on the ratio f by varying the solid conductivity and flow speed. Tables 7 and 8 give all porous systems under study in this thesis at two separate
Reynolds numbers, 1,100 and 2,200 respectively. Nield [2002], by modifying the traditional two-temperature thermal energy equations [Nield & Bejan, 1999], developed a model for porous medium in which the solid material consists of parallel rods, sheets, or parallel tubes. The resulting model yields a simple formula for computing the effective heat conductivity under the condition of local thermal equilibrium,
KKs f (Ks ))eff ==(K f eff (4.7) εεK(s +−1)Kf
Values of the effective heat conductivities of fluid phase obtained from computation at
four axial locations, ∆∆13,,∆4, and ∆5 , for four different solids are listed in both Tables
120 7 and 8, together with the theoretical estimates given by Equation (4.7), as indicated within the parenthesis.
Solid Material Κsolid f @ ∆1 f @ ∆3 f @ ∆4 f @ ∆5
Aluminum 218 1.167 1.159 (1.205) 1.156 (1.205) 1.16
Si3N4 25 1.166 1.18 (1.204) 1.153 (1.204) 1.159
Silicon Rubber 0.166 1.167 1.126 (1.167) 1.127 (1.167) 1.146
Air 0.0262 1.165 1.120 (1.0) 1.116 (1.0) 1.142
Table 7: The ratio of effective heat conductivity of fluid phase, f, at Re = 1,100.
Solid Material Κsolid f @∆1 f @∆3 f @∆4 f @∆5
Aluminum 218 1.166 1.159 (1.205) 1.156 (1.205) 1.16
Si3N4 25 1.166 1.186 (1.204) 1.15 (1.204) 1.16
Silicon Rubber 0.166 1.166 1.170 (1.167) 1.13 (1.167) 1.15
Air 0.0262 1.165 1.137 (1.0 ) 1.124 (1.0 ) 1.145
Table 8: The ratio the effective heat conductivity of fluid phase to inflow heat conductivity, f, at several locations with Re = 2,200.
At location upstream of the porous medium, the resulting averaged heat conductivity of air is 1.165 times that of air at 300K and 1 atm. It is essentially unaffected by doubling the mass flow rate at the inlet, i.e., the Reynolds number. However, a slightly lower magnitude within the porous medium is observed for the hypothetical solid, air. Also, a
121 small increase of Keff in high conducting solid matrices is seen. Overall, this numerically obtained heat conductivity of fluid phase did not show significant changes in its magnitude even for the case in which the heat conductivity of solid phase is 10,000 times larger than that of air. Thus, the observed enhancement of heat transfer in highly conductive solid matrices is not attributable to the heat conductivity of the fluid phase.
4.3.1.3 Thermal Dispersion
The term ‘thermal dispersion’ is used to refer to the “thermal transport enhancement occurring when fluid undergoes mixing as it traverses the tortuous paths around the solid phase in a porous medium” [Calmidi, 1998]. It is the result of the hydrodynamic process involved and the magnitude of thermal dispersion is high compared to molecular diffusion at high Reynolds numbers, especially when the solid phase heat conductivity is low or comparable with its fluid phase counterpart. In the past, several studies as listed in
Table 4 have been attempted to quantify dispersion in the setting of either packed beds or high porosity metal foams. They have provided empirical formulas based on experimental means for prediction of dispersion and the resulting correlated models are applied in their numerical calculations [Calmidi, 1998; Yokoyama, 1995; Hsu, 1990;
Hunt, 1988]. Their numerical approaches treat the dispersion phenomenon by including an additional diffusive term to the stagnant component of fluid phase which can be derived from the energy equation, Equation (1.9), using volume averaging techniques.
The dispersion conductivity Kd is used to express a correlation of fluctuations of velocity and temperature fields. The resulting dispersive conductivity is a tensor having
122 components in streamwise and transverse directions. Vafai [1998] has shown that the effect of longitudinal dispersion is insignificant for P (Péclet number) greater than 10 and the transverse dispersion is much more important because the advection dominates the transport at high Reynolds number. As such, it is documented in the literature that the effect of thermal dispersion is essential in forced convective flows at high Reynolds number and for porous systems with solid phase having conductivity low or comparable to its fluid counterpart.
In the current study, the governing equations do NOT require an externally imposed thermal dispersion model. To quantify the thermal dispersion in forced convection, a hypothetical solid material is theoretically devised so that the interfacial conduction between solid and fluid phases can be excluded. This material is assumed to have the same thermophysical properties as the fluid phase, i.e. air, hence this porous medium is called hypothetical solid/air system in the following analysis. This is very special and unique feature afforded by the numerical approach. This is a powerful thought process that is not easily achievable by experimental or theoretical approach. Firstly, there are enormous difficulties in accessing and making measurements in the limited void (pore) space in porous media. Secondly, it would be difficult, if possible, to find a porous system with the same fluid as the solid phases in real world to perform experiments. To the author’s best knowledge, no study has been carried out at the pore level scale heretofore this thesis.
123 Koch [1985, 1988, 1998] carried out asymptotic analyses in a high porosity medium at arbitrary Péclet numbers, Ρe=PrRe, by using the averaged equations of concentration function. An asymptotic solution of hydrodynamic diffusivity is obtained. In his work, the mechanisms causing dispersion are characterized and their contributions are theoretically predicted. It’s noted that the primary interest of those studies is the macroscopic transport of mass and heat, i.e. transfer on a length-scale larger than that of a detailed microstructure of the porous medium of concern. A brief summary of those studies is given below.
At sufficiently low Péclet number, the molecular diffusion plays the main role of transporting mass and heat through the microstructure because the hydrodynamic mixing is weak as shown later in Figure 4.26. In particular, the effective diffusivity in both transverse and longitudinal directions is independent of Ρe and of the same order as the molecular diffusivity. Therefore, a constant heat transfer coefficient is expected in this regime. In the range of high Péclet number, the bulk convective motion and the mechanical/hydrodynamic dispersion in the transverse direction in porous medium become an important mechanism for transporting mass and heat [Alazmi & Vafai, 2000].
According to Koch and Brady [1985], the effective thermal diffusivity grows linearly with the Péclet number, Ο(Ρe). There are other mechanisms arising from the holdup or stagnant region and boundary layer around solid phases, which respectively give effects of Ο(Ρe2) and Ο(Ρe ℓn Ρe). Also, wall functions are suggested to account for the reduction of mixing near the bounding wall in addition to the change in the velocity profile. [Cheng, 1986: Koch 1998].
124 0.4
Hypothetical solid (air) Re = 1,100 (a) (b) 0.3 (c) (d) (e) Re = 5,500 (a) (b) H 0.2 (c) y/ (d ) (e)
0.1
0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 f
Figure 4.20: Profiles of ratio of heat conductivities f in the transverse direction with respect to two inflow conditions (Re).
In general, the diffusive term in the volume averaged energy equation, Equation 1.9, is modeled by the introduction of thermal dispersion conductivity, Kd, defined to describe the effect of thermal dispersion,
ρCTpu''=⋅Kd∇T (4.8) where u' and T' are the fluctuation of velocity vector and temperature. The thermal dispersion conductivity is a tensor and is sensitive to the direction. In an analogy to turbulence quantities which are obtained by temporal average technique, the porous medium is assumed to be statistically spatial-homogeneous. Thus, the current study can be described analogously as the direct numerical simulation (DNS) in turbulence field.
125 The product term of u'T' can be expressed as terms of u'T', v'T', and w'T' in the axial, transverse, and spanwise directions respectively. Though it is questionable that the porous medium is statistically homogeneous in the transverse direction because of wall effects, the assumption is made to demonstrate the capability of the current research for providing microscopic information to thermal dispersion modeling, in a same manner as the DNS does for the turbulence modeling.
Figure 4.21 plots the dimensionless fluctuating velocity in the transverse direction at the downstream of the hypothetical porous medium, ∆4. The filled symbol indicates the averaged v-velocity, vavg, which is integrated along the axial direction. These plotted results are in nondimensionized by the inflow velocity and height of the channel. It’s observed that the maximum fluctuation is comparable to the inflow velocity which is
0.5m/s (Reynolds number of 5,500). The averaged velocity also fluctuated according to the porous structure but remains within the range ± 0.1m/s.
126 1
0.8 v_avg v-v_avg
0.6 y*
0.4
0.2
0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 v*
Figure 4.21: The dimensionless spatially averaged v-velocity along the longitudinal direction is plotted with the fluctuation, v', deviated from it at the location of ∆4, Re=5,500. Solid symbol denotes the dimensionless spatial average and open symbol denotes the dimensionless fluctuation.
Corresponding to the same location, the temperature fluctuation, T', is shown with the averaged temperature Tavg in Figure 4.22. Tavg is obtained by integrating T along the axial direction. Results are nondimesionized by inflow temperature and the height of channel.
127 1
T_avg 0.8 T-T_avg
0.6 y*
0.4
0.2
0 0 0.5 1 1.5 2 T*
Figure 4.22: The temperature fluctuation, dimensionless T' , is plotted together with the spatially averaged temperature, dimensionless Tavg as indicated in the figure; data are taken at ∆4.
Similarly, the profiles for other cases with Reynolds number ranging from 550 to 2,200 are obtained. An attempt is made to obtain the thermal dispersion conductivity, Kd, from the current study by using the model expressed in Equation (4.8). The resulting Kd is displayed in the following two figures. For the two-dimensional problem under discussion, Figures 4.23 and 4.24 show the product terms of (ρCpu'T')/(dT̃/dx) and
(ρCpv'T')/(dT̃/dy) respectively. They seem completely random, with no recognizable patterns.
128
1
0.8
0.6 y*
Re 0.4 550 810 1,100 2,200 0.2 5,500
0 -1 -0.5 0 0.5 1 1.5 2 2.5 Kd-x
Figure 4.23: The component of Kd in the longitudinal coordinate varies along the height of channel with respect to various flow rate (Reynolds number).
129 1
0.8
0.6 Re 550 y* 810 0.4 1,100 2,200 5,500
0.2
0 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Kd-y
Figure 4.24: The component of Kd in the transverse coordinate varies along the height of channel with respect to various flow rates (Reynolds number).
In order to distinguish the locations where either v' or T' is zero, i.e., either at the solid phases or in the core region, a constant number -1 is placed at these places in Figure 4.24.
The variation of (Kd)y in Figure 4.24 indicates that the dispersion is heterogeneous and the top wall exhibits larger variance with the Reynolds number than the lower wall. In general, the thermal dispersion in the transverse direction is insignificant for low
Reynolds number flow.
130 The overall averaged (Kd)x and (Kd)y are plotted against the Reynolds number in Figure
4.25. It shows different behaviors between (Kd)x and (Kd)y; they are opposite functions of
Reynolds number (or inflow velocity in this study), in which (Kd)x increases with the axial convection velocity. It’s noted that the absolute value of (Kd)y is about the same order of magnitude as the stagnant heat conductivity of fluid phase at higher Reynolds number and can’t be neglected. To verify the above results, additional analysis is conducted in the following which does not follow the same thinking used by the volume averaged method. Also, it is noted that Figure 4.26 present the v' profile at the same location as Figure 4.21 has shown. Both are plotted at the same scale. Reading from both figures together, they confirm the assertion made earlier that the hydrodynamic mixing is insignificant at low Reynolds numbers.
0.14
0.12
0.1 x 0.08 y
0.06
0.04 Kd 0.02
0
-0.02
-0.04
-0.06
0 1000 2000 3000 4000 5000 Re
Figure 4.25: The thermal dispersion conductivity in the x and y direction versus the Reynolds number under case (b) (Equation (4.3)).
131 1
0.8 v_avg v-v_avg
0.6 y*
0.4
0.2
0 -0.6 -0.4 -0.2 0 0.2 0.4 v*
Figure 4.26: The spatially dimensionless averaged v-velocity along the longitudinal direction is plotted with the fluctuation, v', deviated from it at the location of ∆4, Re=550. Solid symbol denotes the dimensionless spatial average and open symbol denotes the dimensionless fluctuation.
To facilitate the discussion and comparison with theoretical approach based on the volume averaging under the same reference viewpoint, the relevant parameters are used: local thermal diffusivity, D=K/(ρCp) as commonly used in fluid dynamics, and Deff(x) which is obtained by averaging D along the transverse direction over the height of the channel. To understand the wall effects on thermal dispersion, computational results of thermal diffusivity D*(x,y) are plotted against y in Figure 4.27 at location ∆3 under conditions of case (a), Equation (4.3) for various Péclet number (or Reynolds number).
132 The non-dimensional diffusivity, D*, is obtained by dividing D by the thermal diffusivity at inflow condition. The thermal dispersive effect seems diminishing outside of the thermal boundary layer, as shown in Figures 4.27 and 4.28, and grows as the wall is approached.
1 Peclet
200.0 0.8 320.0 440.0 600.0 800.0 0.6 1,600 4,000 y*
0.4
0.2
0 0 0.5 1 1.5 2 2.5 3 D*
Figure 4.27: Profiles of non-dimensional diffusivity in the transverse direction with respect to various Péclet number, under wall condition of Case (a) - non symmetric heating.
133
1
Re 550 0.8 810 1,100 2,200 5,500 0.6 y*
0.4
0.2
0 0.5 1 1.5 2 2.5 3 3.5 D*
Figure 4.28: Profiles of non-dimensional diffusivity in the transverse direction with respect to various Re, under wall condition of Case (b) – symmetric heating.
The relationship of D* vs. y* in Figures 4.27 and 4.28 does not agree with the wall function developed by [Koch, 1998] which grows exponentially from 0 at wall to unity at a few particle diameters away from the wall. The discrepancy is attributed to the underlying differences in assumptions. The theoretical results from [Koch, 1998] is an isothermal solution which is based the averaged equations of concentration and coupled with Brinkman like equation for momentum. It is noted that if an analogy between mass and heat transport is to be made, extra care has to be taken to include wall effects because
134 boundary conditions applied to the mass and temperature are different. Contrary to that, the present study is obtained by solving mass, momentum and energy equations together, allowing the conductivity and density to vary with temperature. See Chapter 2 for details.
The effects from higher conductivity and lower density at higher temperature within thermal boundary layer result in a higher heat diffusivity, D = K/(ρCp), in the area closer to wall, as shown in Figure 4.21. This result is physically consistent.
Angirasa & Peterson [1999] concluded that better heat transfer rate can be attained by increasing velocities (or Re). Figure 4.29 displays the factors affecting heat transfer, including Reynolds number (i.e. velocity in this study) , transversely averaged thermal diffusivity, the thermal dispersion conductivity and the hydrodynamic mixing (expressed in terms of vorticity). The Nusselt number is seen to behave as a linear function of
Reynolds number (Péclet number). But, contrary to Koch’s results, the local thermal diffusivity and its mean value in the transverse direction decrease as Reynolds number increases.
135 parameter
30 Nusselt number D*x10 ω∗ x100 25 Kd* in x-direction Kd* in y-direction 20 r e t 15 e m a r 10 pa
5
0
-5 1000 2000 3000 4000 5000 Re
Figure 4.29: Plots of analyses under heating condition of case (a) to summarize thermal dispersion study.
In Figure 30, the effect of non-symmetric heating at both walls on the Nusselt number distributions is displayed where the cooled wall has a lower Nu not only in the smooth region, but also in the porous region.
136
25
one-sided heating 20 two-sided heating
15 Nu
10
5.58 5 4.88 0 0.2 0.4 0.6 0.8 1 x/L
Figure 4.30: Comparison of heat transfer effect in Nusselt number with respect to two heating cases, i.e. case (a) (Equation (4.3)) and case (b) (Equation (4.4)).
4.3.2 Summary
The forced convection is studied for flow in porous media confined between parallel plates on which isothermal conditions are imposed. The remarkable heat transfer enhancement within the porous media is detailed. The overall results indicate that the thermal dispersion is solely determined by the near wall geometric characteristic of the
137 porous medium of concern. The relative importance of this effect is decided by heat conductivity of solid phases. The effective conductivity and thermal dispersion are three- dimensional properties for most porous media which are heterogeneous in practice.
Heterogeneity of porous media is a subject that needs more studies from experimental and computational viewpoints. In addition, there is a need of understanding at the pore- scale parameters in order to provide modeling and parameters at the macroscopic level, for instance, the effective thermal conductivity and thermal dispersion conductivity.
For the highly conductive solid material (aluminum in this study), the thermal dispersion is negligible and the axial conduction dominates the overall heat transfer. The thermal boundary layer is thin, compared with other types of materials. It also confirms that dispersion is relatively important in the overall heat transfer for flow with comparable values of heat conductivity in both fluid and solid phases at higher Reynolds number.
4.4 Three-Dimensional Rectangular Duct flow
4.4.1 Flow with heated porous section
The methodology described in Chapter 2 is extended from a two-dimensional porous structure, like parallel rods or sheets described in the previous section, to a markedly more complex three-dimensional structure, as the real world porous systems. In engineering applications of porous medium, the dimension is often of finite length in that the flow would not reach thermally fully developed. Thus, this three-dimensional study is
138 focused on the developing thermal boundary layer, as occurring in practical applications, for instance in the heat control systems or other energy conversion devices. As shown in
Figure 4.31, a heat source is located right below the porous system as indicated by the darkened line for the isothermal wall of temperature Th. The case presented is a duct flow with a cross-sectional aspect ratio, width to height, of 0.2, and a porous region width of 10% of the channel length. Similar to the two-dimensional channel flow, five sampling locations are selected at which computational results are presented. The numerical setup is shown schematically in Figure 4.31. The air flows from left to right at a predefined inflow conditions of uin = 7.5 m/s (Reynolds number of 32), Tin = Tc =300 K, and density
3 of solid , ρs = 2.01 Kg/m . The porosity of the porous medium in the present three- dimensional case is 0.72. The structure and pores can be seen by plotting the density contours in the next two figures at different viewpoints. During start of this study, computational grids with 80,000 cells were used. The grid is refined until acceptable results are obtained. The final computational grid used contains a total of 400,000 elements, as seen in Figure 4.32. The case considered in this section is a porous system of air and the hypothetical solid of air.
139
Tc
Fully developed flow Flow exit
u , in Porous Medium Tin
y z x ∆1 ∆2 ∆3 ∆4 ∆5
Heated Wall , Th Figure 4.31: Schematic of a three-dimensional porous duct with heat source from below.
Figure 4.32 illustrates both the density and temperature contours along with the mesh used. In the density contours, shown in Figures 4.32 and 4.33 the red color indicates the solid-phase, again displaying its randomness in 3D; the rest of colors represent the density of the fluid phase.
140
Figure 4.32: Contours of temperature and density of the porous system with meshes.
141
Figure 4.33: Contours of density of Y-Z planes in the longitudinal direction of x.
142 The contours of x-component velocity u are given at five cross planes in Figure 4.34.
Figure 4.34: The axial velocity contours at crossplanes (Y-Z) along the longitudinal direction.
Figures 4.35 (a) and (b) provide a vivid axial velocity distribution at mid and exit planes of the porous sample, showing irregular and randomized behavior, which is consistent with what was observed by Okuyama and Abe [2000]. The heat is seen to diffuse in the transverse direction from the bottom heated wall in the temperature contours.
143 (a)
(b)
Figure 4.35: Detailed contours of axial velocity and temperature at the (a) mid-plane (∆3) and (b) downstream (∆4 )of the porous sample.
144 Further downstream from the porous sample, Figure 4.36 indicates heat diffuses more toward the cold wall, because of irregular motion of fluid, with little effect of conduction.
Though the upper right corner remains cold at x* = 0.95, which is close to the exit of the duct, the temperature distribution is fairly homogeneous in most of the duct.
Figure 4.36: Temperature contours in Y-Z planes along the axial direction x.
145 In addition, the profiles of cross plane components at five locations are displayed in
Figures 4.37 and 4.38. The three dimensional nature is evident within the porous region but diminishes near the exit of the duct (x*=0.95). The velocity contours of (u,v,w), given in Figures 4.34, 4.37, and 4.38, at the sampling locations indicate there are areas of accelerating, decelerating and nearly stagnant flow. The velocity profiles at ∆1 (x* =
0.25), are plotted in Figure 4.39, showing a smooth and regular behavior before the flow enters the porous region. Finally, it is noted in Figure 4.40 that the pathlines obtained from four computational rakes at different heights of the duct illustrate the flow swirled around the solid phases and through pores and the pathlines are tortuous, determined by the local pore structure. As a result, the temperature contours at several longitudinal directions indicate that the heat is transferred convectively by the upward motion, see
Figure 4.37 for v contours at the same location, faster at the right corner at the upstream of the porous medium than the rest of area near the heated wall.
146
Figure 4.37: Contours of v, the velocity in the transverse direction of y at five Y-Z planes along the axial direction x.
147
Figure 4.38: Contours of w, the velocity in the transverse direction of y at five Y-Z planes along the axial direction x.
148
Figure 4.39: Velocity contours of three components in X, Y and Z directions at ∆1.
Figure 4.40: Four rakes of pathlines starting at x* = 0.45, upstream of the porous sample, with contours of axial velocity. Each rake consists of 20 particles, located at heights of y* = 0.05, 0.25, 0.5 and 0.75 respectively.
149 4.4.2 Summary
The simulation of a three-dimensional duct with a porous insert has been conducted. A quantitative comparison with experimental data is difficult because scanty measurements available up-to-date. The velocity field in 3D case is a more distorted flow than the 2D one. This distortion is not limited to the near-wall region, as it is in the two-dimensional flow. The temperature field at the exit of the duct shows that the mixing caused by porous insert offers an efficient way to dissipate the heat from the heat sauce. The qualitative description of transport phenomena of flow through a three-dimensional duct is a demonstration of the capabilities of the numerical approach proposed in this thesis.
4.5 Conclusion
The enhancement of heat transfer by the porous media is widely known and observed.
The success of the enhancement is also determined by their abilities to disrupt thermal boundary layer development and to deliver the cold fluid directly into the heated area.
The interplay of conduction, convection, and viscosity within the porous medium determines how the mass and heat being transferred. The proposed numerical approach can provide an improved understanding of related fluid and solid phases interaction, transport phenomena and their effects.
Throughout the whole analysis, conventionally used non-dimensional parameters are adopted in studying forced convection in porous media. They are inherently tied to the
150 overall velocity and temperature fields at flow situations; they don’t really represent the local multiple scale phenomena. Some new parameters might be required, but this not the objective of this thesis.
151 Chapter 5
Closure
5.1 Summary
This thesis proposes a new numerical approach for simulating flow through porous media at pore scale where both fluid and solid phases are taken into account simultaneously. It documents a comprehensive study of transport phenomena in porous media, including flow and forced convective heat transfer in the fluid phase and heat conduction in the solid phase. Various representative materials, commonly seen in engineering applications, are used along with air as the fluid phase.
Different from the widely used averaging methods, this thesis is probably the first effort ever to simulate the transport phenomena in porous media by solving a coupled system of three dimensional Navier-Stokes equations for fluid phase and heat conduction equation for solid phase. In this study, the flow is assumed laminar flow because the Reynolds number based on the pore size is moderate to small, and the solid phase is taken as having constant properties. However the proposed method is not limited by these assumptions.
The only limitation of the present method is by the applicability of the continuum assumption of flow, whose validity is estimated by the Knudsen number. However, in most engineering applications, the pore size is much larger than the mean free path of the
152 fluid of interest and hence the continuum approach is deemed appropriate. While the computation effort is relatively intensive and may be of concern, it is by all means not a limiting factor, considering the power of today’s personal computers and the ever fast growing in speed and memory. Among the unique features of the proposed methodology is that no special treatments are required for the interfacial conditions between solid and fluid phases because of the inherent alignment of solid surface with mesh.
Using the new approach, a numerical study has been conducted which focuses on investigating physical phenomena beyond the macroscopic transport behavior.
Comparison with other work is constrained by the fact that scant data is available in the literature, except some classic aspects of mass and thermal transport in porous media at the macroscopic level. As a result, data obtained in the present study, which is at the pore scale, are averaged and related to relevant non-dimensional parameters. The following major aspects are investigated and summarized below.
In the hydrodynamic aspect, the velocity field and pressure drop are studied based on the structure as well as the properties of solid and fluid phases. The velocity profile exiting the porous medium is non-uniform. Moreover, the resulting velocity distribution which is distorted by the structure of porous matrix affects greatly the heat transfer at bounding walls. The form drag is not only a function of velocity at low speed flow, but also determined by the geometric shape of the solid matrices. It is found that the pressure drop should include the inertia effect at higher Reynolds number and the geometric factor and it grows with the square of velocity.
153 In the heat transfer aspect, conduction and forced convection are studied. The averaged heat conductivity in fluid phase is compared with analytical models for high porosity medium and the current computed result gives excellent agreement. The forced convection analyses are conducted without invoking assumptions commonly used in works by others. The thermal dispersion is studied in conjunction with a devised hypothetical solid material assigned the thermophysical properties of air. The thermal diffusion is mainly controlled by the mixing or vortices occurring in the tortuous passages between solid phases. The thermal diffusivity is found to be a linear function of velocity. It can not be ignored if the fluid and solid phases are having comparable heat conductivity.
5.2 Future work
The proposed methodology is sufficiently flexible that it can be incorporated into other off the shelf codes with reasonable ease since it is only tied with the numerical meshes used in computation, not to the mathematical models or numerical algorithms. Some possible future extensions of interest are suggested in the following.
(1) The transient phenomenon is critical in all energy conversion devices. In order to
gain a better understanding about the operating conditions and morphological
characteristics of porous medium, a systematic study of time accurate simulations
can be conducted.
154 (2) Electrolyte or other catalysts used in the fuel cell involve surface chemistry
between the reactants. This mesh based methodology provides an advantageous
environment for imposing surface reactions at the contacting surface of solid and
fluid phases. The surface reaction mechanism may be specified by the
electrochemistry and neutral multi-step reactions. A balance of reaction and
diffusion can be conveniently enforced at the contacting surfaces of reactant and
reacting fluid because these surfaces are also the computational faces (three-
dimensional mesh) or edges (two-dimensional mesh) of mesh cells.
(3) Surface tension and capillary forces are important for biomedical and space
applications. The proposed methodology can be incorporated into a framework
which is capable of these types of simulations.
(4) In order to improve the computing efficiency, an automatic adaptive mesh
algorithm can be incorporated to improve the numerical accuracy and stability. By
doing so, the mesh is capable of automatically refining itself in regions where
variables changes rapidly and hence a more accurate resolution is necessary.
(5) The structural integrity of solid matrix is of critical concern for the manufacturer
and user. Coupling the present capability with the structural analysis will be
beneficial in investigating the mechanical integrity of porous medium at
operation. In this regard, the analysis probably should consider anisotropy of the
solid phase. Hence, variable properties within the solid should be incorporated for
this application.
155 Appendix A
Random Number Generator
The flowing program is authored by Makoto Matsumoto and Takuji Nishimura [1998].
It’s available and can be downloaded from http://www.math.keio.ac.jp/~matumoto/emt.html.
There are also versions of JAVA and other programming language. Visit the website for more information.
* A C-program for MT19937: Real number version * genrand() generates one pseudorandom real number (double) * which is uniformly distributed on [0,1]-interval, for each * call. sgenrand(seed) set initial values to the working area * of 624 words. Before genrand(), sgenrand(seed) must be * called once. (seed is any 32-bit integer except for 0). * Integer generator is obtained by modifying two lines. * Coded by Takuji Nishimura, considering the suggestions by * Topher Cooper and Marc Rieffel in July-Aug. 1997. * * This library is free software; you can redistribute it and/or * modify it under the terms of the GNU Library General Public * License as published by the Free Software Foundation; either * version 2 of the License, or (at your option) any later * version. * This library is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * See the GNU Library General Public License for more details. * You should have received a copy of the GNU Library General * Public License along with this library; if not, write to the * Free Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA * 02111-1307 USA * * Copyright (C) 1997 Makoto Matsumoto and Takuji Nishimura. * When you use this, send an email to: [email protected] * with an appropriate reference to your work. * *********************************************************************** * * Fortran translation by Hiroshi Takano. Jan. 13, 1999. * * genrand() -> double precision function grnd() * sgenrand(seed) -> subroutine sgrnd(seed) * integer seed
156 * * This program uses the following non-standard intrinsics. * ishft(i,n): If n>0, shifts bits in i by n positions to left. * If n<0, shifts bits in i by n positions to right. * iand (i,j): Performs logical AND on corresponding bits of i and j. * ior (i,j): Performs inclusive OR on corresponding bits of i and j. * ieor (i,j): Performs exclusive OR on corresponding bits of i and j. * *********************************************************************** * * this main() outputs first 1000 generated numbers program main * implicit integer(i-n) implicit double precision(a-h,o-z) * parameter(no=1000) dimension r(0:7) * * call sgrnd(4357) * any nonzero integer can be used as a seed do 1000 j=0,no-1 r(mod(j,8))=grnd() if(mod(j,8).eq.7) then write(*,'(8(f8.6,'' ''))') (r(k),k=0,7) else if(j.eq.no-1) then write(*,'(8(f8.6,'' ''))') (r(k),k=0,mod(no-1,8)) endif 1000 continue * stop end *********************************************************************** * subroutine sgrnd(seed) * implicit integer(a-z) * * Period parameters parameter(N = 624) * dimension mt(0:N-1) * the array for the state vector common /block/mti,mt save /block/ * * setting initial seeds to mt[N] using * the generator Line 25 of Table 1 in * [KNUTH 1981, The Art of Computer Programming * Vol. 2 (2nd Ed.), pp102] * mt(0)= iand(seed,-1) do 1000 mti=1,N-1 mt(mti) = iand(69069 * mt(mti-1),-1) 1000 continue * return
157 end *********************************************************************** * double precision function grnd() * implicit integer(a-z) * * Period parameters parameter(N = 624) parameter(N1 = N+1) parameter(M = 397) parameter(MATA = -1727483681) * constant vector a parameter(UMASK = -2147483648) * most significant w-r bits parameter(LMASK = 2147483647) * least significant r bits * Tempering parameters parameter(TMASKB= -1658038656) parameter(TMASKC= -272236544) * dimension mt(0:N-1) * the array for the state vector common /block/mti,mt save /block/ data mti/N1/ * mti==N+1 means mt[N] is not initialized * dimension mag01(0:1) data mag01/0, MATA/ save mag01 * mag01(x) = x * MATA for x=0,1 * TSHFTU(y)=ishft(y,-11) TSHFTS(y)=ishft(y,7) TSHFTT(y)=ishft(y,15) TSHFTL(y)=ishft(y,-18) * if(mti.ge.N) then * generate N words at one time if(mti.eq.N+1) then * if sgrnd() has not been called, call sgrnd(4357) * a default initial seed is used endif * do 1000 kk=0,N-M-1 y=ior(iand(mt(kk),UMASK),iand(mt(kk+1),LMASK)) mt(kk)=ieor(ieor(mt(kk+M),ishft(y,-1)),mag01(iand(y,1))) 1000 continue do 1100 kk=N-M,N-2 y=ior(iand(mt(kk),UMASK),iand(mt(kk+1),LMASK)) mt(kk)=ieor(ieor(mt(kk+(M-N)),ishft(y,- 1)),mag01(iand(y,1))) 1100 continue y=ior(iand(mt(N-1),UMASK),iand(mt(0),LMASK)) mt(N-1)=ieor(ieor(mt(M-1),ishft(y,-1)),mag01(iand(y,1)))
158 mti = 0 endif * y=mt(mti) mti=mti+1 y=ieor(y,TSHFTU(y)) y=ieor(y,iand(TSHFTS(y),TMASKB)) y=ieor(y,iand(TSHFTT(y),TMASKC)) y=ieor(y,TSHFTL(y)) * if(y.lt.0) then grnd=(dble(y)+2.0d0**32)/(2.0d0**32-1.0d0) else grnd=dble(y)/(2.0d0**32-1.0d0) endif * return end
159 Appendix B
Governing Equations and Numerical Method
This appendix gives a complete description of the conservation equations and the numerical methods used for studying transport phenomena of porous flows.
B.1 Governing Equations
The objects under consideration consist of both fluid and solid phases. Thus, the conservation laws governing the transport phenomena in each of these phases will be included and solved simultaneously.
In the fluid phase, air is considered and its major species, such as N2 and O2, are included in the solution with an intent for practical applications to flows in porous media. The governing equations are the usual conservation equations of mass, momentum, and energy, but they are listed below for completeness. For the solid phase, the only relevant equation is the energy conservation. First, the fluid phase equations are expressed in vector form, ∂∂Q ∂ ∂ ++()E-E (F-F )+(G-G )=S (B.1) ∂∂tx vv∂y ∂z v where the vector Q contains the conservative variables, the vectors E, F, and G are the inviscid fluxes, and the corresponding viscous fluxes are denoted by the subscript “v”. These vectors are: Q = (ρρ, uv, ρ, ρwH, ρ , ρY, ρY,..., ρY )T tN12 s −1 E =+(ρρuu, 2 p, ρuv, ρuw, ρuH, ρuY, ρuY,..., ρuY )T tN12 s −1 F =+(ρρρv, v, v2 p, ρvw, ρvH , ρvY , ρvY ,..., ρvY )T tN12 s −1 G =+(ρρρwww, , v, ρw2 p, ρwH, ρwY, ρwY,..., ρwY )T tN12 s −1 T E = (0,ττ, ,τ,uτ++vwττ+q,ξ,ξ,...,ξ) (B.2) v xx xy xz xx xy xz x x12x x Ns −1 F =+(0,ττ, ,τ,uvτ τ+wτ+q,ξ,ξ,...,ξ )T v yx yy yz yx yy yz y y12y y Ns −1 G =+(0,ττ, ,τ,uvτ τ+wτ+q,ξ,ξ,...,ξ )T v zx zy zz zx zy zz z z12z z Ns −1 S = 0
160 The source term S vanishes because no radiation, body forces, chemical reactions, or 22 2 phase change is considered. The total entahlpy is Ht = hu++(v+w)/2 and the total energy EHtt=−p/ ρ . The notation for the rest of variables are rather standard and their meanings are listed in nomentclature. The Cartesian components of the viscous stress tensor, heat flux vector, diffusion mass flux vector of each species are expressed as
∂uu1 ∂∂v∂w τµ=−2 [ ( ++)], xx ∂∂xx3 ∂y∂z ∂∂uv ττ==µ()+, (B.3) xy yx ∂∂yx ∂∂uw ττ==µ()+ xz yz ∂∂zx ∂T q =−K (B.4) x ∂x ∂Y ξρ= D i (B.5) xi mi ∂x
where Dmi is the diffusion coefficient of species i within the gas mixture. Noticing that only the x-components are given here, the counterparts in the y and z directions are simply permutated.
Air is assumed to be an ideal mixture of Ns species following the ideal gas law,
Ns (B.6) p = ρRTu∑Yi/ mi i=1
where mi is the molecular weight of species i and Ru is the universal gas constant. The mixture enthalpy h is evaluated by weighting in terms of species mass fraction Yi and enthalpy,
Ns (B.7) h= ∑Yihi i=1 where T hT()=+h C ()TdT (B.8) ifi∫ pi Tref
161 For species i, the heat capacityC is evaluated by a 4th-degree polynomial, h is the pi f i heat of formation of species i at Tref . The viscosity and heat conductivity coefficients for the gas mixture are evaluated by weighting with molar fraction Xi,
Ns (B.9) µ = ∑ X iµi i=1 and 1 NNii (B.10) K(=∑∑XiiK+Xi/Ki) 2 ii==11
rd The species properties, µi and κi are given by 3 -order polynomials in temperature. Solid regions are interspersed within the fluid domain and can be properly described by the unsteady heat conduction equation. No internal heat source is assumed inside the solid regions. The temperature range is relatively low so that radiative heat transfer is negligible. Therefore the energy equation in the solid resuces to the heat conduction equation :
∂T ρ Cs =∇i(K ∇T) (B.11) s v∂t ss
where the subscript “s” denotes the solid phase and Cv is the heat capacity of the solid.
To close the system of partial differential equations, boundary conditions are imposed. At the solid surfaces, either the interfaces between the fluid and solid matrix or the solid walls bounding the flow domain, the usual no-slip conditions are appropriate for the velocity components and temperature. Hence, fluid velocity vanishes and temperatures of fluid and solid are equal at their interfaces. Also, the adiabatic wall condition may be employed at the bounding walls. In addition, inflow and outflow conditions are imposed whereever appropriate.
B.2 Numerical Method
The above set of conservation equations are valid for describing compressible flows over a wide range of Mach numbers, from very low to hypersonic (since chemical reactions are allowed in the formulation). However, the flow studied here lies entirely in the low Mach regime, or nearly incompressible flow. Numerically, there arise two main difficulties that render compressible flow algorithms ineffective at low Mach numbers, namely: (1) the round-off error caused by the pressure gradient term in the momentum
162 equations (the pressure term is of order 1/ M 2 larger than the convective term in the momentum equations as M → 0 ) and (2) the stiffness caused by the wide disparities in the eigenvalues associated with the invsicd flow system (the largest ratio of eigenvalues is O(1/M)). To circumvent these two problems related to low Mach number simulations, standard techniques can be prescribed. First, the static pressure p is decomposed into a constant reference pressure pref and a gauge pressure pg which is then used as a working variable in the solution. Secondly, a pseudo-time preconditioning term is added to rescale the inviscid-system eigenvalues. Applying these two procedures to the gas-phase equations results in the following enhanced set of equations.
∂∂qQ∂ ∂ ∂ Γ ++()E-E +(F-F )+(G-G )=S (B.12) ∂∂τ tx∂ vv∂y ∂z v
It is noted that a pseudo time τ is introduced only for numerical purposes, in order to distinguish from the physical time t. The new variable vector q and the associated pre- conditioning matrix Γ are given as
q = ( pu, ,v, w,h,Y,Y,...,Y )T (B.13) g12 Ns −1
⎡⎤1/ β 0 iiiiiii0 ⎢⎥iiiiii ⎢⎥u /0βρ 0 ⎢⎥v /0βρ0iiiii0 ⎢⎥ ⎢⎥w/0βρ00iiii0 ⎢⎥ Ht /1βρ− 000 0iii0 Γ = ⎢⎥ (B.14) ⎢⎥Y1 /0βρiii 0ii0 ⎢⎥Y /βρiiii000i ⎢⎥2 ⎢⎥i iiiii00ρ 0 ⎢⎥i iiiiii00ρ ⎢⎥ ⎢⎥Y /0β iiiiii0ρ ⎣⎦Ns −1
In finite-volume discretization, the fluxes at the cell faces are defined via various numerical methods to satisfy the conservation equations. Now the preconditioned system, Eq. (B.12), is discretized in space using a cell-centered scheme for an unstructured mesh, as illustrated in Fig. A.1 for a 2D triangular grid. A set of ordinary differential equations are obtained for unknowns, qi , i =1, 2,…,Nc, over Nc cells.
−1 ∂q Γ T ii=− Rq( ), q=(qq, ,...,q ) (B.15) i12 Nc ∂∆τ Vi
163 where Ri(q) is the residual expressing the balance of fluxes (inviscid and viscous terms) flowing in and out of the cell i. It is given as a sum of numerical flues,
∂Q Ni() Rqii()=∆VA( )i+∑{(E-Evv)kx,,k+(F-F)kAyk ∂t (B.16) k =1 +−()G-Gv kzAA,k Dk(q)}|k|
where Ni() is the number of cells (control volumes) contributing to the changes of qi .
(A ,A ) x y inb(2)
k = 3 inb(3) k = 2 ∆V i k = 1
inb(1)
Figure B.1. 2D finite volume cells.
The subscript “k” denotes the average value at the cell face k whose unit normal vector is given by (Ax, Ay) and the cell volume is denoted by ∆V .
Intensive research efforts have been devoted in the last decades to finding reliable (accurate and stable) and efficient numerical fluxes. In general, the numerical fluxes functions can be broadly classified into two categories, namely centered and upwind schemes.
164 Since a centered scheme for the first derivative terms is employed, a numerical dissipation term denoted by Dk (q) must be added in Eq. (B.16) to mitigate odd-even oscillations and numerical instability. The method proposed by Jameson [1981] is adopted, which contains both the second and fourth difference terms whose coefficients are denoted by ε (2) and ε (4) respectively,
(2) (4) (B.17) Dkn()q=−fac(k){ε [(qqi b()k) (i)]+ε [d2q(inb()k)−d2q(i)]}
fac()k=Rface()k [ris (nb()k )+ris ()] (B.18)
Nifaces () 1−mm1 (B.19) R face()k =={(λλmax kr)}, s (i) ∑ {max(l)} Nifaces () l=1
2 Njfaces () d2q()j=∑ q(l)−q(j) (B.20) Njfaces () l=1
The factor fac(k) is determined by a measure of spectral radii ri(). The indices associated with the cell i are defined in Fig. B.1. The exponent m is a user-specified number. Nfaces ( j) is the number of faces surrounding the cell j.
The maximum eigenvalue used in Eq. (B.19) is associated with the Jacobian matrix in the direction normal to the cell face,
−1 ∂EF∂∂G A =+Γ (AA+A) (B.21) nx∂∂qqy∂qz where ( AA, , A) are the Cartesian components of the unit normal vector of the cell xyz face. The largest eigenvalue of A is found to be determined by velocity V , area vector n A , speed of sound c, and the parameter β ,
1 ββ2 222 λβmax =+{}||VA[1 ]+VA(1−)+4(AAAxxx++) (B.22) 2 cc22 where VVA ==iAuAxy+vA+wAz. The parameter β appearing in the preconditioning matrix is chosen to render all eigenvalues of An same order of magnitude and is defined by
22 2 2 2 2 ββ=0 min {max(VVir, ef)}, V=u+v+w (B.23) ∀∈iN[1, s ]
165 where β0 is a constant of O(1) and Vref is a reference velocity usually chosen to be on the order of the free stream velocity. The coefficients for the second and fourth difference terms are specified by the user and they were fixed at -0.01 and 0.04 respectively throughout this study.
For steady-state flows, the first-order differential equation in pseudo-time τ is solved by the following explicit 4-stage Runge- Kutta scheme,
(0) n qqii==, i 1,2,3,... −1 (1) (0) ∆τ iiΓ 0 qqii=−[(Riq)] 4 ∆Vi −1 (2) (0) ∆τ iiΓ (1) qqii=−[(Riq)] 3 ∆Vi −1 (B.24) (3) (0) ∆τ iiΓ (2) qqii=−[(Riq)] 2 ∆Vi −1 (4) (0) ∆τ iiΓ (3) qqii=−[(R iq)] 1 ∆Vi n+1(4) qqii= where the superscript n represents the previous pseudo-time level, n+1 represents the new pseudo-time level, and the superscripts (1), (2), (3), and (4) denote the stages of the Runge-Kutta scheme. The unsteady term is set to zero in the pseudo-time residual .
The pseudo-time step for each cell is calculated from
CFLi∆V ∆=τ i (B.25) i Nifaces () ∑ λmax ()k k =1
where the parameter CFL is a user-specified value and set to 2.0 throughout the study. The same pseudo-time step is used for all the gas-phase equations.
The solution of solid phase is performed in the same way as the fluid phase, except that it is even simpler because only one scalar equation is involved. Coupling between the fluid and solid phases are accomplished through the imposing of boundary conditions in which
166 the temperature and heat flux are enforced to be continuous at each interface between the fluid and solid cells.
167 Appendix C
The computed pressure drop for test cases of forced convective heat transfer. The uint used is in N/m2(Pa).
Substrate material as inner plate as external plate Silicon Nitride 4.e-4 1.96e-4 Aluminum 6.1e-4 2.79e-4
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